Composite Finite Elements and Multigrid · 2009-09-08 · A multigrid method for the eﬃcient...

Composite Finite Elements

and Multigrid

Dissertation

zur

Erlangung der naturwissenschaftlichen Doktorwurde(Dr.sc.nat.)

vorgelegt der

Mathematisch-naturwissenschaftlichen Fakultat

der

Universitat Zurich

von

Nadin Stahn

aus

Deutschland

Begutachtet von

Prof.Dr. Stefan Sauter

Prof.Dr. Wolfgang Hackbusch

Zurich 2006

Die vorliegende Arbeit wurde von der Mathematisch-naturwissenschaftlichen Fakultat derUniversitat Zurich auf Antrag von Prof.Dr. Stefan Sauter und Prof.Dr. Michel Chipotals Dissertation angenommen.

ii

Contents

Abstract v

Zusammenfassung vii

1 Introduction 1

1.1 Model Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Galerkin Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Standard Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Linear Elements for Ω = (a, b) . . . . . . . . . . . . . . . . . . . . . . 5

1.3.2 Linear Elements for Ω ⊂ R2 . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Error Estimates for Finite Elements . . . . . . . . . . . . . . . . . . . . . . 10

1.4.1 Error Estimates with respect to H1(Ω) . . . . . . . . . . . . . . . . . 10

1.4.2 Error Estimates in L2(Ω) . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Composite Finite Elements 13

2.1 Concept of the Composite Finite Element Method . . . . . . . . . . . . . . 13

2.2 Coarse Scale Discretisations . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Grid Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.2 Composite Finite Element Hierarchy . . . . . . . . . . . . . . . . . . 15

3 Multigrid Method 19

3.1 Intergrid Transfer Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Coarse Grid Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Multigrid Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.1 Twogrid Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.2 Abstract Multigrid Algorithm . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4.1 Twogrid Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.2 Multigrid Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Numerical Experiments 27

4.1 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Model Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2.1 2D Model Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2.2 1D Model Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

iii

5 Convergence Analysis for the 1D Model Problem 395.1 Proof of the Smoothing Property . . . . . . . . . . . . . . . . . . . . . . . . 405.2 Proof of the Approximation Property . . . . . . . . . . . . . . . . . . . . . . 455.3 Multigrid Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6 Convergence Analysis for the 2D Model Problem 496.1 Splitting of the Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.1.1 Properties of the Single Block Matrices . . . . . . . . . . . . . . . . 516.1.2 Analysis of the Mass Matrix . . . . . . . . . . . . . . . . . . . . . . . 53

6.2 Proof of the Smoothing Property . . . . . . . . . . . . . . . . . . . . . . . . 576.2.1 Estimates of ‖WI‖0←0, ‖WII‖0←0 and ‖I − ωWI‖0←0 . . . . . . . . 596.2.2 Estimate of ‖WI(I − ωWI)

ν‖0←0 . . . . . . . . . . . . . . . . . . . 606.3 Proof of the Approximation Property . . . . . . . . . . . . . . . . . . . . . . 65

6.3.1 Estimate of ‖VI‖0←0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.3.2 Partitioning of the system matrix . . . . . . . . . . . . . . . . . . . . 706.3.3 Estimate of ‖VII‖0←0 . . . . . . . . . . . . . . . . . . . . . . . . . . 716.3.4 Estimate of ‖VIII‖0←0 . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.4 Multigrid Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7 Summary 85

A Basics 87A.1 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87A.2 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89A.3 Domains and Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90A.4 Functional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92A.5 Theorems for Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 93

B Numerical Tests 95

C Notation 99

iv

Abstract

A multigrid method for the efficient solution of elliptic boundary value problems on com-plicated domains will be presented and analysed.The efficiency of multigrid algorithms is based on a multiscale discretisation of the bound-ary value problem. In many practical applications coarse scale discretisations are notavailable because the minimal resolution of the geometry by a finite element grid mightcontain a large number of cells. To overcome this restriction we employ composite finiteelements for the construction of the sequence of coarse-level discretisations. The minimaldimension of the coarsest linear system is very small, independent of the number and sizeof geometrical details in the domain and at the boundary. Numerical results demonstratethe efficiency of the proposed multigrid method.The main objective of this work is the investigation of the convergence of the multigrid al-gorithm based on composite finite element discretisations. The idea is to adapt the generalconvergence theory for geometric multigrid to our specific situation and to prove the so-called smoothing and approximation property. The proof of these properties differs fromthe standard situation for the following reason: The discretisation is based on overlappingtriangulations where the integration is carried out on the intersections of the triangleswith the domain. Since these intersections may have quite arbitrary shapes and, possibly,very tiny measures the multigrid theory for shape regular meshes cannot be applied. Thestandard regularity theory is not applicable as well due to the, possibly, degenerate inter-section of triangles with the domain. However, by using weighted norms for the smoothingand approximation property we can prove that the multigrid algorithm converges robustlywith respect to the smallness of these intersections. Numerical examples illustrate thisrobust convergence behaviour.

v

vi

Zusammenfassung

In dieser Arbeit wird eine Mehrgittermethode zur effizienten Losung von elliptischenRandwertproblemen vorgestellt und analysiert. Die Effizienz von Mehrgitteralgorithmenbasiert auf einer mehrskaligen Diskretisierung der Randwertaufgabe. In vielen praktischenAnwendungen sind solche grobskaligen Diskretisierungen nicht verfugbar, da die minimaleAuflosung der zugrundeliegenden Geometrie mittels herkommlicher Diskretisierungsver-fahren, wie der Finite-Elemente-Methode zu einer großen Zahl von Unbekannten fuhrt.Zur Bewaltigung dieser Restriktion werden zusammengesetzte Finite Elemente eingefuhrt,mit welchen man eine Folge von grobskaligen Diskretisierungen konstruiert. Dabei ist dieDimension des Gleichungssystems, welches aus der grobsten Diskretisierung hervorgeht,sehr klein und unabhangig von der Anzahl und Große geometrischer Details im Gebiet undam Rand des Gebietes. Numerische Ergebnisse verdeutlichen die Effizienz der vorgestelltenMehrgittermethode.Der Hauptbestandteil dieser Arbeit ist die Konvergenzanalyse des Mehrgitteralgorithmusbasierend auf Diskretisierungen mit zusammengesetzten Finiten Elementen. Dazu passenwir die Konvergenztheorie der geometrischen Mehrgittermethode an unsere spezielle Situ-ation an und beweisen die sogenannte Gattungs- und Approximationseigenschaft. Der Be-weis dieser Eigenschaften unterscheidet sich signifikant vom Standardfall aufgrund folgen-der Schwierigkeit: Die Diskretisierung mit zusammengesetzten Finiten Elementen basiertauf uberlappenden Triangulierungen. Das heißt, das Integrationsgebiet, welches die Schnitt-menge eines Elements mit der offenen Menge, auf der die Randwertaufgabe formuliert ist,darstellt, kann ein sehr kleines Maß haben. Dieses fuhrt zu unterschiedlichen Skalierun-gen in der Systemmatrix. Aufgrund dieses Skalierungseffektes ist die standard Regu-laritatstheorie nicht anwendbar. Unter Zuhilfenahme gewichteter Normen beweisen wir dieGlattungs- und die Approximationseigenschaften hinsichtlich der Kleinheit des Uberlapps.

vii

viii

1 Introduction

The numerical solution of boundary value problems is very important in various fields ofphysics and engineering sciences. In many practical applications as, e.g., environmentalmodelling or simulation of engines the underlying physical objects have extremely compli-cated shape containing a huge number of small geometric details. This thesis is concernedwith efficient numerical solutions of partial differential equations on domains with com-plicated geometric structures. These geometric details are, for instance, holes or a roughboundary. In this work we restrict our attention to elliptic boundary value problems. Thefinite element method is a method for the numerical solution of differential equations.The resolution of the geometric structures in the domain by a conventional finite elementmesh leads to a huge dimension of the system of discrete equations. This system will besolved by using the multigrid method. The efficiency of multigrid algorithms is based ona multiscale discretisation of the problem. However, in many practical applications coarsescale discretisations are not available due the following reasons:

• The minimal resolution of the geometry by a finite element grid might contain ahuge number of cells.

• A fine scale discretisation based on an unstructured grid is given and no grid hier-archy for coarse scale discretisations is available.

This drawback was the motivation of the development of so-called composite finite ele-ments. These elements allow coarse scale discretisations of problems with complicatedgeometries by only few unknowns, the minimal number being independent of the num-ber and size of the geometric details. In this work a robust multigrid method, based oncomposite finite element discretisations, for solving elliptic boundary value problems oncomplicated domains is presented. The convergence analysis is given for a one-dimensionaland a two-dimensional model problem.In this chapter a general model problem will be introduced (c.f. Section 1.1). The mathe-matical treatment of the finite element algorithms is based on the variational formulationof elliptic differential equations. The concept of the finite element method is formulatedin Section 1.3. In Chapter 2 we will introduce composite finite elements. In [HS97a],[HS97c] and [Sau97] composite finite elements are developed for the approximation ofpartial differential equations on complicated domains. These finite elements allow coarsescale discretisations where the minimal number of unknowns does not depend on the shapeof the domain. In [SW04] and [War03] composite finite elements are developed for thediscretisation of elliptic boundary value problems with discontinuous coefficients. The def-inition of the composite finite elements is based on a hierarchical coarsening of standardfinite elements which fits in the concept of the multigrid method.In Chapter 3, we present a multigrid algorithm for the efficient solution of the arisinglarge system of linear equations. This algorithm contains features from both, geometric

1

2 1 Introduction

multigrid (c.f. [Hac85]) and algebraic multigrid methods (c.f. [RS87]). In [FHSW03],the algorithmic aspects of this multigrid method have been presented. The developmentof multigrid methods for partial differential equations on complicated domains is a topicof vivid research. Various different approaches exist in the literature as, e.g., algebraicmultilevel methods (c.f. [RS87], [VMB96], [Bra86], [MBV99]), coarsening via geometricagglomeration ([BX95], [BX96], [CXZ98]), and coarsening by auxiliary space methods(see [Xu96], [GN98]). For a detailed comparison of the different approaches we refer to[BHW98].Chapter 4 is concerned with numerical experiments. These numerical results motivate thedevelopment of special model problems in two dimensions and one dimension for studyingthe convergence of the presented multigrid algorithm.

In Chapter 5 and Chapter 6 the convergence analysis for the two model problems is workedout explicitely. The convergence is proved in the framework of geometric multigrid method,i.e., by investigating the so-called smoothing and approximation property. The proof ofthese properties differs for composite finite element discretisations on overlapping gridssignificantly from the situation considered in [Hac85] for the following reason: Trianglesat the boundary, possibly, have only small overlap with the physical domain. The robustconvergence of the multigrid algorithm is investigated with respect to these boundaryeffects.

1.1 Model Problem

First we will introduce the model problem. Let Ω ⊂ Rd be a bounded domain with

Lipschitz boundary Γ := ∂Ω (see A.12). We assume that Γ is partitioned into two disjoint,open and measurable subsets ΓD, ΓN such that Γ = ΓD ∪ ΓN. As a prototype for an ellipticboundary value problem of second order we consider the Poisson equation

−∆u = f in Ω . (1.1)

We consider mixed boundary conditions

u = 0 on ΓD

∂u

∂n= 0 on ΓN

homogeneous Dirichlet boundary condition and

homogeneous Neumann boundary condition. (1.2)

Assumption 1.1 Let the Dirichlet part of the boundary ΓD be non-empty.

To formulate the variational equivalent of (1.1), we define an energy space that incorporatesthe essential, i.e., Dirichlet, part of the boundary condition. Let

X0 :=

u ∈ C∞(Ω) :

√∑

|t|≤1

‖Dtu‖2L2(Ω)

<∞

.

The completion of X0 in L2(Ω) with respect to the norm

‖u‖H1(Ω) :=

√∑

|t|≤1

‖Dtu‖2L2(Ω)

1.1 Model Problem 3

results in the Sobolev space H1(Ω) (see Definition A.15). We define the energy spaceH1

D(Ω) as

H1D(Ω) := u ∈ H1(Ω) : u|ΓD

= 0 , (1.3)

where u|ΓD= 0 is understood in the sense of traces. The appropriate bilinear form for

the variational problem is determined by multiplying Poisson’s equation by test functions,integrating over Ω and then integrating by parts. We define the bilinear form a : H1

D(Ω)×H1

D(Ω) → R by

a(u, v) :=

∫

Ω

〈∇u,∇v〉dx . (1.4)

Weak formulation of Poisson’s equation 1.2 Let f ∈ L2(Ω). We are seeking a func-tion u ∈ H1

D(Ω) such that

a(u, v) =

∫

Ω

fv dx ∀v ∈ H1D(Ω) . (1.5)

Proposition 1.3 Any solution of (1.1) solves (1.5). If the solution of (1.5) is in H2(Ω),then it solves (1.1) almost everywhere.

The bilinear form a(·, ·) is symmetric. Existence and uniqueness of a solution of problem(1.5) follow from the continuity and coercivity of a: Let u, v ∈ C∞0 (Ω). The continuityfollows via the Schwarz’ inequality.

|a(u, v)| =

∣∣∣∣∣∣

∫

Ω

〈∇u,∇v〉dx

∣∣∣∣∣∣

=∣∣(u, v)H1(Ω)

∣∣ ≤ (u, u)

1/2H1(Ω)

(v, v)1/2H1(Ω)

= ‖u‖H1(Ω)‖v‖H1(Ω)

with the continuity constant CΩ = 1. Since C∞0 (Ω) is dense in H10 (Ω) (see Theorem

A.19), a(·, ·) has an extension to H1D(Ω) × H1

D(Ω) and is bounded by the same constant(c.f. Lemma A.20). For all u ∈ H1

D(Ω), Friedrichs’ inequality A.18 implies: There exists aconstant Ccoerc depending only on the domain Ω such that

a(u, u) ≥ Ccoerc‖u‖2H1(Ω) .

Therefore, a(·, ·) is coercive on H1D(Ω) (see Definition A.5). The existence and the unique-

ness of a solution of problem (1.5) follow directly from the Lax-Milgram Theorem A.7.

Proposition 1.4 Problem (1.5) has a unique solution.

The finite element method allows the approximation of the solution of problem (1.5).The resolution of the geometric structures in the domain leads to a high dimension ofthe system of discrete equations. The system of discrete equations will be solved byusing the multigrid method, because this method solves elliptic boundary value problemswith optimal complexity. The efficiency of multigrid methods is based on a multiscalediscretisation of the problem. Composite finite elements allow coarse scale discretisationsof problems with complicated geometries by only few unknowns, the minimal number beingindependent of the number and size of the geometric details. We recall the definition ofthe standard finite elements in Section 1.3 and introduce the composite finite elements inChapter 2.

4 1 Introduction

1.2 Galerkin Discretisation

We approximate the solution of the problem (1.5) by solutions of discrete, finite dimen-sional problems. For this purpose we apply the Galerkin discretisation method and replacethe space H1

D(Ω) by a finite dimensional subspace S(Ω) ⊂ H1D(Ω). This leads to the fol-

lowing discrete problem.

Problem 1.5 Let f ∈ L2(Ω) be given. Find uS ∈ S(Ω) such that

a(uS , v) =

∫

Ω

fv dx ∀v ∈ S(Ω) .

The existence and uniqueness of a solution of this problem follow again by the Lax-MilgramTheorem A.7, because S(Ω) is a finite dimensional subspace of H1

D(Ω).

Proposition 1.6 Problem 1.5 has a unique solution.

Next we write Problem 1.5 in terms of a basis of S(Ω). Let ϕi : 1 ≤ i ≤ n be a basis ofS(Ω), where n is the dimension of S(Ω). Problem 1.5 is equivalent to seeking a functionuS ∈ S(Ω) which satisfies

a(uS , ϕi) = f(ϕi) ∀i = 1, . . . , n . (1.6)

Because uS ∈ S(Ω) there exists u = (ui)ni=1 ∈ R

n with

uS =n∑

i=1

uiϕi . (1.7)

Plugging this ansatz into (1.6) results in a system of linear equations for the coefficientvector u.

n∑

j=1

uja(ϕj , ϕi) = a(uS , ϕi) = f(ϕi) ∀i = 1, . . . , n .

Remark 1.7 Equation (1.7) introduce the mapping

Pu =

n∑

i=1

uiϕi

from Rn onto S(Ω). Note that P is an isomorphism.

Let (L)ij = a(ϕj , ϕi) and (f)i = f(ϕi) =∫

Ω fϕi dx for i, j = 1, . . . , n. Then Problem 1.5is equivalent to solving

Lu = f . (1.8)

for given f ∈ L2(Ω). The matrix L is called system matrix. Let u be the solution to thevariational problem (1.5) and uS be the solution to the approximation Problem 1.5. Wecan estimate the error ‖u− uS‖ by using the following theorem.

1.3 Standard Finite Elements 5

Theorem 1.8 (Cea) Suppose V (Ω) is a closed subspace of the Hilbert space H1(Ω) anda(·, ·) is a continuous, coercive bilinear form on V (Ω). For the discrete Problem 1.5 wehave

‖u− uS‖V (Ω) ≤CΩ

Ccoercminv∈S(Ω)

‖u− v‖V (Ω) ,

where CΩ is the continuity constant and Ccoerc is the coercivity constant of a(·, ·) on V (Ω)(see Definition A.5).

For the proof we refer, e.g., to [BS94, theorem (2.8.1)]. Cea’s Theorem shows that uS isquasi-optimal in the sense that the error ‖u − uS‖V (Ω) is proportional to the error of thebest approximation up to a multiplicative constant.

1.3 Standard Finite Elements

In the finite element method, the physical domain can be discretised into a number ofuniform or non-uniform geometric finite elements that are connected via nodes (see SectionA.3). The unknown solution is approximated within each element by a finite elementfunction. The finite element function is characterised by its values at the nodes. Theresults of this discretisation method can be reformulated into a matrix equation of theform Lu = f (see (1.8), which is subsequently solved for the unknown variable.We restrict ourselves to the consideration of linear finite elements. For the introductionof finite elements of higher order we refer, e.g., to [BS94] or [Bra97].The basis ϕi : 1 ≤ i ≤ n is commonly chosen so that the stiffness matrix L =(Lij)1≤i,j≤n = (a(ϕj , ϕi))1≤i,j≤n is sparse. Let the bilinear form a(·, ·) be given by(1.4). Let Bi denote the interior of the support supp (ϕi) of the basis function ϕi, i.e.,Bi := supp (ϕi) \ ∂ supp (ϕi). A sufficient condition that ensures Lij = a(ϕj , ϕi) = 0 isBi ∩Bj = ∅, because the integration

a(ϕj , ϕi) =

∫

Ω

〈∇ϕj ,∇ϕi〉

can be restricted to Bi ∩ Bj. In order to satisfy this condition the basis functions shouldhave small supports.

1.3.1 Linear Elements for Ω = (a, b)

Let us first investigate the one-dimensional boundary value problem

−u′′(x) = g(x) for a < x < b , u(a) = u(b) = 0 . (1.9)

Assume there is a partition of the interval (a, b) given by a = x0 < x1 < · · · < xn+1 = b.Denote the subintervals by τi := (xi−1, xi) for all 1 ≤ i ≤ n + 1. For the subspaceSn(Ω) = Sn((a, b)) ⊂ H1

0 ((a, b)) the piecewise linear functions are chosen by:

Sn(Ω) := u ∈ C0([a, b]) : u|τi is affine for all 1 ≤ i ≤ n+ 1 , u(a) = u(b) = 0 .(1.10)

The continuity u ∈ C0([a, b]) is equivalent to the continuity at the nodes xi, that meansfor all 1 ≤ i ≤ n+ 1 it holds u(xi + 0) = u(xi − 0).

6 1 Introduction

Remark 1.9 u ∈ Sn(Ω) is uniquely determined by its nodal values u(xi) with 1 ≤ i ≤n+ 1:

u(x) =u(xi)(xi+1 − x) + u(xi+1)(x− xi)

xi+1 − xifor x ∈ τi+1 ,

where u(x0) = u(xn+1) = 0. Because of this boundary condition one concludes thatSn(Ω) ⊂ H1

0 (Ω). The (weak) derivative u′ ∈ L2(Ω) is piecewise constant:

u′(x) =u(xi+1) − u(xi)

xi+1 − xifor x ∈ τi+1 .

The basis functions ϕi are given by (see Figure 1.1)

ϕi(x) :=

(x− xi−1)/(xi − xi−1) for xi−1 < x ≤ xi ,(xi+1 − x)/(xi+1 − xi) for xi ≤ x ≤ xi+1 , ,0 otherwise.

1 ≤ i ≤ n

r r r r r r............................................................................................................................................................................................................................................................................................................................................................................................................................

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.

x0 = a x1 x2 xn−1 xn b = xn+1

ϕ2

ϕ1 ϕn

ϕn−1

Figure 1.1: Basis functions of the linear finite elements on (a, b).

One has the representation u(x) =∑n

i=1 u(xi)ϕi. The supports of the basis functions ϕiare τ i ∪ τ i+1 = [xi−1, xi+1].

The weak formulation of the boundary value problem (1.9) is

a(u, v) = f(v) (1.11)

with

a(u, v) :=

b∫

a

u′v′ dx , f(v) :=

b∫

a

gv dx u, v ∈ H10 (Ω) .

The Galerkin discretisation of (1.11) via finite elements is given by: Find uS ∈ Sn(Ω) suchthat

a(uS , v) = f(v) ∀v ∈ Sn(Ω) . (1.12)


The basis representation of (1.12) leads to the linear system Lu = f . For the matrixelements Lij it holds Lij = 0 for all |i− j| ≥ 2. For |i− j| ≤ 1 one obtains

Li,i−1 = a(ϕi−1, ϕi) =

xi∫

xi−1

−1

xi − xi−1

1

xi − xi−1dx =

−1

xi − xi−1,

Lii = a(ϕi, ϕi) =

xi∫

xi−1

1

(xi − xi−1)2dx+

xi+1∫

xi

1

(xi+1 − xi)2dx =

1

xi − xi−1+

1

xi+1 − xi,

Li,i+1 = a(ϕi, ϕi+1) =

xi+1∫

xi

−1

(xi+1 − xi)2dx =

−1

xi+1 − xi.

The right-hand side f = ((f)i)1≤i≤n is given by

(f)i = f(ϕi) =

xi+1∫

xi−1

gϕi dx

=1

xi − xi−1

xi∫

xi−1

g(x)(x− xi−1) dx+1

xi+1 − xi

xi+1∫

xi

g(x)(xi+1 − x) dx .

The resulting system of equations Lu = f is tridiagonal, thus, in particular, sparse.

Remark 1.10 Let f ∈ H10 (Ω)′ and u ∈ H1

0 (Ω) be the solution of problem (1.11). Let Ibe the Interpolation on Sn(Ω), that means

I : H1(Ω) → Sn(Ω) ; u 7→n∑

i=1

u(xi)ϕi(xi) .

Then Iu is the solution of the discrete problem (1.12):

uS = Iu .

If we have mixed boundary conditions in (1.9), we can write w.l.o.g. u(b) = 0 (seeAssumption 1.1), then V = H1

D(Ω). The subspace Sn ⊂ H1D(Ω) results from (1.10) after

removal of the condition u(a) = 0. In order that dimSn = n the numbering has to bechanged: The partition of Ω = (a, b) is given by a = x1 < x2 < x3 < · · · < xn+1 = b.

1.3.2 Linear Elements for Ω ⊂ R2

We assume Ω ⊂ R2 is a polygon. This polygon is divided into triangles τi.

Definition 1.11 T := τ1, τ2, . . . , τk is called a conforming triangulation of Ω if thefollowing conditions are fulfilled:

1. τi, 1 ≤ i ≤ k, are open triangles, so-called finite elements,

2. τi are disjoint, i.e., τi ∩ τj = ∅ for i 6= j,

3.⋃

1≤i≤k τ i = Ω,

8 1 Introduction

4. for i 6= j there holds either

(a) τ i ∩ τ j = ∅ , or

(b) τ i ∩ τ j is a common vertex of the elements τi and τj , or

(c) τ i ∩ τ j is a common edge of τi and τj.

Let T be a conforming triangulation. The point x is called a node of T if x is a vertex ofone finite element τi ∈ T .

Remark 1.12 Condition 4. in Definition 1.11 avoids hanging nodes.

One distinguishes interior and boundary nodes, according to whether x ∈ Ω or x ∈ ∂Ω.Let n be the total number of interior nodes. We define the finite element space Sn(Ω) asthe subspace of piecewise linear functions:

Sn(Ω) := u ∈ C0(Ω) : u = 0 on ∂Ω , u(x) = ai0 + ai1x1 + ai2x2 on τi ∀1 ≤ i ≤ n .(1.13)

That means, on each triangle τi ∈ T the function u is a linear function. Since u = 0 on∂Ω it holds Sn(Ω) ⊂ H1

0 (Ω). The dimension of Sn(Ω) is the number of interior nodes ofthe triangulation T . Each function u ∈ Sn(Ω) is uniquely determined by its nodal valuesu(xi) at the interior nodes xi, 1 ≤ i ≤ n.

Remark 1.13 Let xi, 1 ≤ i ≤ n, be the interior nodes of the triangulation T . For anarbitrary ui, 1 ≤ i ≤ n, there exists exactly one u ∈ Sn(Ω) with u(xi) = ui. It is of theform u =

∑ni=1 uiϕi, where the basis functions ϕi are characterised by

ϕi(xi) = 1 , ϕi(xj) = 0 for j 6= i . (1.14)

If τ ∈ T is a triangle with the vertices xi = (xi1, xi2), xℓ = ((xℓ1, xℓ2) and xm =((xm1, xm2), then

ϕi(x) =(x1 − xℓ1)(xm2 − xℓ2) − (x2 − xℓ2)(xm1 − xℓ1)

(xi1 − xℓ1)(xm2 − xℓ2) − (xi2 − xℓ2)(xm1 − xℓ1)on τ. (1.15)

On any τ ∈ T that do not have xi as a vertex, ϕi ≡ 0.

Using (1.15) we get a linear function defined on all τ ∈ T , which satisfies (1.14). Theproperty (1.14) forces the continuity at all nodes. If the vertices xj = (xj1, xj2) andxk = (xk1, xk2) are directly connected by the edge of a triangle τ ∈ T , then (1.15) yieldsthe representation

ϕi(xj + s(xk − xj)) = sϕi(xk) + (1 − s)ϕi(xj) with s ∈ [0, 1]

for both triangles that have this side in common. Thus ϕi is also continuous on thecommon edges, so that ϕi ∈ C0(Ω). Applying this consideration to two boundary nodesxℓ and xm with ϕi(xℓ) = ϕi(xm) = 0 we derive ϕi = 0 on ∂Ω, and thus ϕi belongs toSn(Ω).

The support of the basis function ϕi is

supp (ϕi) =⋃

τ : τ ∈ T has xi as a corner .


ζ

η

(0, 0)

(1, 0)

(0, 1)

τ∗

φ

φ−1

x

y

x1

x2

x3

τ

Figure 1.2: Transformation of triangle τ to the reference triangle τ∗.

The bilinear form associated to the Poisson problem is given by a(u, v) =∫

Ω〈∇u,∇v〉dx(see (1.4)). The integrals

Lij = a(ϕj , ϕi) =∑

k∈N

∫

τk

〈ϕj , ϕi〉dx

where N := k ∈ N : τk ∈ T has xi and xj as vertices.We have Lij = 0 if xi and xj are not directly connected by the edge of a triangle. Foreach 1 ≤ k ≤ n one can express the integral over the triangle τk ∈ T as an integral overthe reference triangle τ∗ in Figure 1.2.Let xi = (xi, yi), i = 1, 2, 3, be the vertices of triangle τ ∈ T , and let τ∗ be the unittriangle in Figure 1.2. Then, for φ : τ∗ → τ it holds:

• φ : (ζ, η) 7→ x1 + ζ(x2 − x1) + η(x3 − x1) maps τ∗ onto τ .

• detφ′(ζ, η) = |(x2 − x1)(y3 − y1) − (y2 − y1)(x3 − x1)| = 2|τ | for all ζ, η ∈ R.

• Thus,∫

τ

v(x,y) dxdy = 2|τ |∫

τ∗

v(φ(ζ, η)) dζ dη .

If one replace the Dirichlet condition u = 0 on ∂Ω by mixed boundary conditions u = 0on ΓD and ∂u/∂n = 0 on ΓN with Γ = ΓD ∪ ΓN, then the following changes take place:

1. n = dimSn is the number of all inner nodes and all boundary nodes xi ∈ ΓN.

2. Sn(Ω) ⊂ H1D(Ω) is given by (1.13) with the restriction u = 0 on ΓD instead of ∂Ω.

In general we will use the following notation:

Definition 1.14 Let

S(Ω) := v ∈ C0(Ω) : v|τ is affine for all τ ∈ T , v = 0 on ΓD ,

the space of piecewise continuous linear finite elements with zero-boundary conditions atΓD. In the sequel, we will denote these finite elements standard finite elements. Note, that

10 1 Introduction

ΓD 6= ∅ by Assumption 1.1. Let ϕxx∈Θ be the nodal basis of S(Ω), i.e., for all x ∈ Θ isϕx ∈ S(Ω) and

ϕx(y) = δxy . (1.16)

Note that this basis is a Lagrange-basis.

1.4 Error Estimates for Finite Elements

We restrict ourselves to the consideration of the linear finite elements from the last sectionand therefore we assume: T is a conforming triangulation of Ω ⊂ R

2. We recall the resultsof error estimates for the finite element method and refer for the proofs of the followinglemmata and theorems to a textbook of Hackbusch [Hac92, Section 8.4].

1.4.1 Error Estimates with respect to H1(Ω)

In order to have a good approximation uS of the solution u of Problem 1.5 one has todetermine d(u, S(Ω)) := min‖u− v‖H1(Ω) : v ∈ S(Ω) (see Theorem 1.8). To do this onebegins by looking at the reference triangle from Figure 1.2.

Theorem 1.15 Let α0 be the smallest interior angle of all τi ∈ T , while h is the maximumlength of the edges of all τi ∈ T . Then

infv∈S(Ω)

‖u− v‖Hk(Ω) ≤ C(α0)h2−k‖u‖H2(Ω)

for k = 0, 1 and all u ∈ H2(Ω) ∩H1D(Ω).

We now consider a sequence of triangulations Ti, i ∈ N. In the construction of thesetriangulations it should be noted that with increasing refinement, hi → 0 as i tends toinfinity, the interior angle may not also be reduced. That means, we assume a sequence ofquasi-uniform triangulations (see Definition A.11). The case k = 1 of Theorem 1.15 maynow be written as follows:

Theorem 1.16 Let Ti, i ∈ N, be a sequence of quasi-uniform triangulations. Then thereexists a constant C, such that for all h = hi and S(Ω) = Si(Ω) the following estimateholds:

infv∈S(Ω)

‖u− v‖H1(Ω) ≤ Ch‖u‖H2(Ω)

for all u ∈ H2(Ω) ∩H1D(Ω).

The combination of this theorem with Theorem 1.8 leads to:

Theorem 1.17 Let Ti, i ∈ N, be a sequence of quasi-uniform triangulations and let thebilinear form a(·, ·) be continuous and satisfy

infsup|a(u, v)| : v ∈ Si(Ω), ‖v‖H1(Ω) = 1 : u ∈ Si(Ω), ‖u‖H1(Ω) = 1 = Ci > 0 .

(1.17)

1.4 Error Estimates for Finite Elements 11

Suppose the constants Ci are bounded below by Ci ≥ Cinf−sup > 0. Let Problem 1.5 havethe solution u ∈ H2(Ω) ∩ H1

D(Ω). Let uS ∈ Si(Ω) be the finite-element solution. Thenthere exists a constant C, which is independent of u, hi, and i, such that

‖u− uS‖H1(Ω) ≤ Ch‖u‖H2(Ω) .

If the assumption u ∈ H2(Ω) ∩H1D(Ω) is replaced by u ∈ Hs(Ω) ∩H1

D(Ω) with 1 ≤ s ≤ 2,and Ω is assumed to be sufficiently smooth, then there holds

‖u− uS‖H1(Ω) ≤ Chs−1‖u‖Hs(Ω) . (1.18)

Remark 1.18 The ellipticity of the bilinear form a(·, ·) replaces the weaker inf-sup con-dition (1.17).

The first estimate in Theorem 1.17 gives the order of convergence O(h), but it requires thatthe solution u ∈ H1

D(Ω) lies in H2(Ω). This is unrealistic for problems e.g. with reentrantcorners while (1.18) holds always for some s > 3/2. The second estimate requires theweaker assumption u ∈ Hs(Ω) ∩H1

D(Ω) with s ∈ [1, 2].

1.4.2 Error Estimates in L2(Ω)

We now consider error estimates for u−uS in the L2-norm. To estimate ‖u−uS‖L2(Ω), weuse a duality argument. The adjoint problem equals Problem 1.5 since a(·, ·) is symmetric:Find u ∈ H1

D(Ω) with

a∗(u, v) = f(v) ∀v ∈ H1D(Ω) , (1.19)

which uses the adjoint bilinear form a∗(u, v) := a(v, u). We say that (1.19) has H2-regularity if: For each f ∈ L2(Ω) the problem (1.19) has a solution u ∈ H2(Ω) ∩H1

D(Ω)with

‖u‖H2(Ω) ≤ Cr‖f‖L2(Ω) . (1.20)

The following error estimates with respect to L2(Ω) result from the Aubin-Nitsche-LemmaA.8.

Theorem 1.19 Let the regularity condition (1.20) be satisfied. Let S(Ω) be the space oflinear finite elements of a conforming and quasi-uniform triangulation. Assume the bilinearform a(·, ·) is continuous and fulfils the inf-sup-condition (1.17) with C ≥ Cinf−sup > 0,and

infv∈S(Ω)

‖u− v‖H1(Ω) ≤ C0h‖u‖H2(Ω) ∀u ∈ H2(Ω) ∩H1D(Ω) .

Let Problem 1.5 have the solution u ∈ H1D(Ω). Let uS ∈ S(Ω) ⊂ H1

D(Ω) be the finite-element solution. Then, with a constant C1 independent of u and h,

‖u− uS‖L2(Ω) ≤ C1h‖u‖H1(Ω) .

If the solution u belongs to H2(Ω) ∩H1D(Ω), then there is a constant C2, independent of

u and h, such that

‖u− uS‖L2(Ω) ≤ C2h2‖u‖H2(Ω) .

12 1 Introduction

Elliptic boundary value problems like (1.5) can be treated by standard finite elements. Butthe condition that the triangulation T has to resolve the boundary of the domain makes acoarse-scale discretisation of problems on complicated domains very costly. The compositefinite elements introduced in [HS97c] allow coarse-level discretisations of partial differentialequations, where the minimal number of unknowns is independent of the number and sizeof geometrical details in the domain Ω.

2 Composite Finite Elements

In this chapter we will introduce composite finite elements. In contrast to standard finiteelements, the minimal dimension of the approximation space is independent of the domaingeometry and this is especially advantageous for problems on domains with complicatedmicro-structures.

2.1 Concept of the Composite Finite Element Method

The principal idea is that these new finite elements allow to adapt hierarchically theshape of the finite element function to the characteristic behaviour of the solution. Theconstruction of the composite finite elements allows to define a hierarchy of discretisationsfor the variational problem (1.5). The definition of composite finite elements is based ona sequence of grids. In contrast to standard finite element grids, only the finest grid hasto resolve the boundary. The definition of composite finite element grids consists of twosteps:

1. In the first step, a hierarchy of auxiliary grids will be defined (c.f. Section 2.2.1).The domain Ω has to be contained in the domain covered by the grids but it is notrequired that the boundary Γ is resolved.

2. In the second step, we remove all triangles having empty intersection with the do-main.

On the finest grid, the composite finite element space coincides with the usual finiteelement space. All lower-dimensional spaces are subspaces of the fine grid space.Let G = τi : 1 ≤ i ≤ N be a shape-regular triangulation (see Definition A.11) of anoverlapping domain Ω∗ ⊃ Ω. Here, and in the sequel, Θ denotes the set of mesh points inG and ♯Θ = n. Let S(Ω∗) denote the standard finite element space for Ω∗ (c.f. Definition1.14)

S(Ω∗) := v ∈ C0(Ω∗) : v|τ is affine for all τ ∈ G, v = 0 on ΓD

The composite finite element space on the domain Ω is defined as the restriction

SCFE(Ω) := S(Ω∗)|Ω := u|Ω : u ∈ S(Ω∗) .

The Galerkin discretisation of Problem 1.5 via composite finite elements is given by: FinduG ∈ SCFE(Ω) such that

a(uG , v) =

∫

Ω

fv dx ∀v ∈ SCFE(Ω) . (2.1)

13

14 2 Composite Finite Elements

As a basis for the space SCFE(Ω) we choose the restrictions (ϕi(x)|Ω)x∈Θ of the standardnodal basis ϕi(x) for the space S(Ω∗). The shapes of two typical basis functions of com-posite finite element spaces are depicted in Figure 2.1. The basis functions equal the nodalbasis functions where the support is restricted to the domain Ω. The basis representation

Figure 2.1: Basis functions for problems with Neumann boundary conditions (left) andDirichlet boundary conditions (right).

of (2.1) leads to the linear system

Lu = f , (2.2)

where the matrix L ∈ Rn×n and the right-hand side f ∈ R

n are given by

(L)ij = a (ϕj |Ω, ϕi|Ω) and (f)i =

∫

Ω

fϕi(x)|Ω dx .

The solution of the system of linear equation (2.2) is linked to the solution of (2.1) via

uG =

n∑

i=1

(u)iϕi|Ω .

2.2 Coarse Scale Discretisations

In this section, we will explain a method for the generation of a hierarchy of coarse scalediscretisations. We give two variants for the generation. The first variant requires a givenfine grid G arose from the discretisation of the boundary value problem via finite elements.This given fine grid resolves the geometry of the domain Ω. But in most applications onehas no fine scale discretisation on hand and therefore we restrict our attention in the sequelon the second variant of generating a hierarchy of coarse scale discretisations.

2.2.1 Grid Hierarchy

Variant 1: This subsection is devoted to the construction of a hierarchy of coarse scalegrids (Gk)kmax

k=0 from a given fine grid G. These grids Gk will be nested for 0 ≤ k ≤ kmax but,possibly, do not resolve the geometric details of the domain Ω. However, the constructionguarantees that the grid Gkmax has a similar (slightly coarser) distribution of mesh cells

2.2 Coarse Scale Discretisations 15

as G. Let Ω0 denote an axis-parallel bounding box of Ω and G0 = τ0, τ1, . . . , τℓ0 aminimal conforming triangulation of Ω0. The finite element grid G0 has the property, thatall elements τi have non-zero intersection with the domain Ω. The grid covers Ω but itis not resolving the Neumann boundary ΓN. The minimal possible number of elementsis small independent of the size and number of geometric details in the domain. Finergrids Gk are generated by recursively refining the initial grid G0. The refinement processis controlled by the distribution of mesh cells in G. A mesh cell τ ∈ Gk is marked forrefinement, if more than three distinct elements of G are contained in τ . Since, we applyconforming subdivisions of elements as, e.g., connecting midpoints of edges, either τi ⊂ τjor (τi ∩ τj = ∅ holds for any τj ∈ Gk and τi ∈ Gk+1, k+ 1 ≤ kmax. To avoid hanging nodesthe green-closure algorithm (cf.[BS81]) is applied. The refinement stops if all mesh cellscontain at most three triangles of the given grid. Triangles having no intersection withthe domain are removed from these grids. For a detailed description of the algorithm andnumerical experiments, we refer to [FHSW03].

Remark 2.1 The algorithm can be generalised to three-dimensional problems in a straight-forward manner.

Variant 2: For the construction of a hierarchy of overlapping grids we start with an initialgrid G0 = τ1, τ2, . . . , τN. By connecting midpoints of the edges of the elements of coarsertriangulations we generate the next finer grid. Triangles having no intersection with thedomain are removed from these grids. Then, (Gk)0≤k≤kmax

is a sequence of overlappinggrids. The grids Gk are nested for 0 ≤ k ≤ kmax but, in general, they do not resolve theboundary Γ.

In both variants the auxiliary grids Gk have the following property:

Remark 2.2 The auxiliary grids are nested in the sense that, for all τ ∈ Gk and k ≤ kmax,the set of sons is contained in τ

⋃

τ∈ sons(τ)

τ ⊂ τ . (2.3)

We assume that Ω =⋃

τ∈G τ and introduce, for 0 ≤ k ≤ kmax, the domains covered by thegrids Gk by

Ωk :=⋃

τ∈Gk

τ .

Θk denotes the set of all vertices of the triangles in Gk. The number of unknowns isdenoted by nk. The definition of auxiliary grids (Gk)kmax

k=0 implies for all 0 ≤ k ≤ kmax

Ω ⊂ Ωk+1 ⊂ Ωk and Θk+1 ⊂ Ωk .

2.2.2 Composite Finite Element Hierarchy

For the formulation of the multigrid algorithm we assume that the Dirichlet part of theboundary ΓD is exactly resolved by the boundary of the union of some triangles τk,i ofthe grid Gk for all 0 ≤ k ≤ kmax. Composite finite elements differ from standard finiteelements only in a vicinity of non-resolved geometric structures. Since the Dirichlet part isresolved we restrict to the discretisation of boundary value problems with Neumann-type


boundary conditions. For the discretisation of pure Dirichlet problems or mixed boundaryvalue problems we refer to [HS97b] and [RSS05].Variant 1: Let Sk(Ωk) = S(Ωk) (respectively S(Ω)) denote the standard finite elementspace on the Ωk (respectively on Ω) given by Definition 1.14. In view of the nestednessof the domains Ω ⊂ Ωkmax ⊂ Ωkmax−1 ⊂ · · · ⊂ Ω0, we introduce interpolation operatorsIkmax : Skmax(Ωkmax) → S(Ω) and, for 1 ≤ k ≤ kmax, operators Ik,k−1 : Sk−1(Ωk−1) →Sk(Ωk) by

Ikmaxu :=

n∑

i=1

u(xi)ϕi and Ik,k−1u :=

nk∑

i=1

u(xk,i)ϕk,i . (2.4)

For 0 ≤ k ≤ kmax, the composition of the interpolation operator over several grids yieldsthe iterated interpolation Ik : Sk(Ωk) → S(Ω)

Ik := Ikmax Ikmax,kmax−1 Ikmax−1,kmax−2 · · · Ik+1,k .

Definition 2.3 The composite finite element space SCFEk (Ω) is the range of Sk(Ωk) under

the mapping Ik

SCFEk (Ω) := Iku : u ∈ Sk(Ωk) .

Remark 2.4 The definition of the composite finite element spaces implies the nestedness

SCFE0 (Ω) ⊂ SCFE

1 (Ω) ⊂ · · · ⊂ SCFEkmax−1(Ω) ⊂ SCFE

kmax(Ω) ⊂ S(Ω) .

The inclusion (2.3) implies

Ik,k−1u = u|Ωk∀u ∈ Sk−1(Ω) . (2.5)

Since the interpolation Ikmax : SCFEkmax

(Ωkmax) → S(Ω) is on non-nested grids Gkmax and G,property (2.5) does not hold in general for Ikmax . Instead, we have

Iku = Ikmax

(u|Ωkmax

)∀u ∈ Sk(Ωk) .

Hence, a finite element function in SCFEk (Ω) is not affine on triangles of Gk but composed of

continuous, piecewise linear pieces on triangles of G. This property motivates the notationcomposite finite elements. This discretisation method for elliptic boundary problems withNeumann boundary conditions was first introduced by Hackbusch and Sauter in [HS97c].

In this thesis another kind of composite finite element hierarchy is used.

Variant 2: The main assumption of the following construction is also the nestedness ofthe domains Ω ⊂ Ωkmax ⊂ Ωkmax−1 ⊂ · · · ⊂ Ω0. The general idea is already mentioned inthe introduction of this section (cf. page 13). The composite finite element space SCFE

k (Ω)is the restriction of the standard finite element space on the overlapping grid Gk to thedomain Ω,

SCFEk (Ω) := S(Ωk)|Ω := u|Ω : u ∈ S(Ωk) .

For each level 0 ≤ k ≤ kmax, the discretisation of (1.5) is given by seeking uk ∈ SCFEk such

that

a(uk, vk) =

∫

Ω

fvk dx , ∀ vk ∈ SCFEk (Ω) . (2.6)

2.2 Coarse Scale Discretisations 17

As a basis of SCFEk (Ω) we choose the usual nodal basis ϕk,i(x)x∈Θk

(hat functions) onΩk restricted to the domain Ω. For each level 0 ≤ k ≤ kmax, the basis representation of(2.6) leads to the linear system

Lkuk = fk , (2.7)

where the matrix Lk ∈ Rnk×nk and the right-hand side fk ∈ R

nk are given by

(Lk)ij = a (ϕk,j|Ω, ϕk,i|Ω) and (fk)i =

∫

Ω

fϕk,i(x)|Ω dx .

The solution of the system of linear equation (2.7) is linked to the solution of (2.6) via

uk =

nk∑

i=1

(uk)iϕk,i|Ω .

This equation defines an isomorphism Pk : Rnk → SCFE

k (Ω)

uk = Pkuk . (2.8)

The adjoint operator P ∗k : Sk(Ω) → Rnk is defined by

〈P ∗k u,v〉 = (u, Pkv)L2(Ω) , (2.9)

with the scalar product

〈u,v〉 =

nk∑

i=1

(u)i(v)i for coefficient vectors u,v ∈ Rnk . (2.10)

We define the grid-dependent norm

‖u‖0 := 〈u,u〉1/2 ∀u ∈ Rnk . (2.11)

Equipped with ‖·‖0 the composite finite element space SCFEk (Ω) becomes a Banach space,

where

‖u‖0 := ‖u‖0,Ω :=

∫

Ω

|u(x)|2 dx

1/2

∀u ∈ SCFEk (Ω) . (2.12)

We skip the index Ω if there is no ambiguity.Since the composite finite element spaces SCFE

k (Ω) are restrictions of standard finite ele-ment spaces one can use the error estimates for standard finite elements given in Section1.4. The existence of an extension operator can be derived from

Theorem 2.5 (Stein) Let Ω ⊂ Ω∗ ⊂ Rd be bounded Lipschitz domains. Then there

exists a continuous extension operator EStein : Hk(Ω) → Hk(Ω∗).

For the proof we refer to [Ste70].


3 Multigrid Method

The multigrid method allows solving elliptic boundary value problems like e.g. (1.5) withoptimal complexity. Under mild conditions, each iteration step of the multigrid methodhas a complexity which is linear in the number of unknowns. Each iteration step of amultigrid algorithm reduces the discretisation error by a constant independent of hk. If,additionally, the convergence rate of the multigrid algorithm is bounded below away from1 independent of the level k, the system of linear equations

Lkuk = fk

can be solved with linear complexity up to a given precision.The multigrid method is the combination of the application of a classical solver, whichis called smoother, and a coarse grid correction. The smoothing step has the effect ofdamping out the highly oscillating part of the error. The smooth part of the error canthen accurately be corrected on the coarser grid.The efficiency of multigrid algorithms strongly relies on a given hierarchy of coarse leveldiscretisations. However, in the case of very complicated domains such a hierarchy is notavailable in an obvious way. Here, composite finite elements will be applied for constructingappropriate coarse-level systems, see Chapter 2. The parameter k ∈ N with 0 ≤ k ≤ kmax

describes the discretisation level. The number of levels kmax is not known a priori. Thekey ingredients of the multigrid method are:

• a hierarchy of discretisations given by the composite finite elements on a hierarchyof grids, c.f. Chapter 2,

• prolongation and restriction operators pk,k−1, rk−1,k between the discretisations, and

• smoothing operators Kk.

The contents of this chapter is based on the books of Hackbusch [Hac85] and Brenner/Scott[BS94, Chapter 6].

3.1 Intergrid Transfer Operators

In order to map coarse grid functions to fine grid functions we need intergrid transfer opera-tors, the prolongation operators. The bijection Pk, defined in (2.8), gives rise to a canonicalchoice of the prolongation pk,k−1 : R

nk−1 → Rnk and the restriction rk−1,k : R

nk → Rnk−1.

Let vk−1 ∈ Rnk−1. The corresponding finite element function is v := Pk−1vk−1. We write

vk = pk,k−1vk−1, or equivalently v = Pkvk ∈ SCFEk (Ω). The canonical choice is v = v

using the inclusion SCFEk−1 (Ω) ⊂ SCFE

k (Ω). Hence, pk,k−1vk−1 is the coefficient vector of

v = v with respect to the basis of SCFEk (Ω). Since P−1

k exists on SCFEk (Ω), we have the

following definition.

19

20 3 Multigrid Method

Definition 3.1 The canonical choice of pk,k−1 : Rnk−1 → R

nk is

pk,k−1 := P−1k Pk−1 .

The canonical restriction rk−1,k : Rnk → R

nk−1 is its transposed rk−1,k = p⊺

k,k−1,

rk−1,k := Rk−1R−1k ,

where Rk = P ∗k is the adjoint of Pk defined by (2.9).

Remark 3.2 The intergrid transfer operators pk,k−1 : Rnk−1 → R

nk and rk−1,k : Rnk →

Rnk−1 are equivalently characterised by

Pk−1 = Pkpk,k−1 and rk−1,kRk = Rk−1 . (3.1)

The following remark concerns the sparsity pattern of pk,k−1. Due to the nestedness ofthe grids Gk, for all 0 < k ≤ kmax, all fine grid points xk ∈ Θk are covered by the coarsegrid mesh Gk−1.

Definition 3.3 For a nodal point xk,i of the grid Gk, we associate a coarser triangleτk−1(xk,i) ∈ Gk−1 by the condition

xk,i ∈ τk−1(xk,i) . (3.2)

If this choice is non-unique, we fix one of all possible triangles.

Remark 3.4 The prolongation matrix pk,k−1 is sparse in the sense

(pk,k−1)ij 6= 0 ⇐⇒ xk−1,j is a vertex of τk−1(xk,i) .

Thus, a matrix row in pk,k−1 contains at most three non-zero elements (pk,k−1)ij ∈0, 1/2, 1.

Hence, the prolongation operator pk,k−1 can be realised locally by first associating toany point xk,i one coarse grid triangle τk−1(xk,i) satisfying (3.2) and secondly linking thevertices of τk−1(xk,i) to xk,i and writing the prolongation weight

(pk,k−1)ij = ϕk,i(xk−1,j) for all vertices xk−1,j of τk−1(xk,i)

into this link where ϕk,i denote the standard basis function of Sk(Ωk).

3.2 Coarse Grid Matrices

Let L : H1D(Ω) →

(H1

D(Ω))′

be the operator associated with the bilinear form a(·, ·) definedin (1.4)

a(u, v) = (Lu, v) ∀u, v ∈ H1D(Ω) ,

where (·, ·) denotes the duality pairing between (H1D(Ω))′ ×H1

D(Ω). Let 〈·, ·〉 be the Eu-

clidean scalar product on Sk(Ωk) × Sk(Ωk) (see (2.10) and let Rk :(H1

D(Ω))′ → Sk(Ω) be

the adjoint of Pk defined in (2.8)

〈Rku,v〉 = (u, Pkv)L2(Ω) for v ∈ Rnk and u ∈ Sk(Ωk) . (3.3)

3.3 Multigrid Algorithm 21

Lemma 3.5 The system matrix Lk and the right hand side fk of the discrete problemLkuk = fk (c.f. 1.8) on the discretisation level k satisfy

Lk = RkLPk and fk = Rkf . (3.4)

The proof is given in [Hac85, p. 47].

Lemma 3.6 Let Lk and Lk−1 be the system matrices with respect to the discretisationlevels k and k − 1. Let the intergrid transfer operators pk,k−1 and rk−1,k be canonicaldefined by (3.1). Then it holds

Lk−1 = rk−1,kLkpk,k−1 . (3.5)

Proof. Equation (3.4) yields Lk = RkLPk and Lk−1 = Rk−1LPk−1. Then equations (3.1)imply

Lk−1 := (rk−1,kRk)L(Pkpk,k−1) = rk−1,k(RkLPk)pk,k−1 = rk−1,kLkpk,k−1 .

The coarse grid matrix Lk−1, that means the system matrix with respect to the nextcoarser grid, is given by (3.5). This definition is called the Galerkin product since equa-tion (3.5) holds for the stiffness matrices of the Galerkin method (c.f. Section 1.2). Thedefinition of the coarse grid matrix implies that, for all 0 ≤ k ≤ kmax the matrix Lkis sparse. This sparsity is very important for the efficiency of the two- or the multigriditeration, since otherwise the smoothing step becomes increasingly expensive (c.f. Subsec-tion 3.3.1). Another advantage of the Galerkin approach is that, if Lk is symmetric andpositive definite, then Lk−1 is also symmetric and positive definite, since rk−1,k = p⊺

k,k−1.For a further discussion of this method we refer to [Hac85, Section 3.7].

3.3 Multigrid Algorithm

The multigrid algorithm is an iterative solver. For each level k = 0, . . . , kmax we assume aniteration method for solving the equation Lkuk = fk on SCFE

k (Ω) is specified. The twogridscheme has two main features: smoothing, that means damping out the high oscillatingpart of the error, on SCFE

k (Ω) and error correction on SCFEk−1 (Ω). In order to define the

multigrid algorithm one has to employ the iterative solver on each level. In the following,we restrict to linear solvers. Exemplarily, we apply here the damped Jacobi iteration withdamping factor ω ∈ (0, 1) as a smoother. Let Dk denote the diagonal part of Lk. Then,one iteration step is given by

u(i+1)k := u

(i)k − ωD−1

k

(

Lku(i)k − fk

)

. (3.6)

The ν-fold application of this solver defines the mapping K(ν)k

(

u(i)k , fk

)

:= u(i+ν)k .

3.3.1 Twogrid Algorithm

In a first phase, ν steps of the iterative solver (smoothing iteration) (3.6) are appliedyielding an approximation uk of uk. The iteration matrix of the smoother is given by

Kk := I − ωD−1k Lk . (3.7)


Denote the approximation error by ek := uk − uk and observe

Lkek = Lkuk − fk =: dk . (3.8)

The idea of the coarse grid correction is to exploit the smoothness of the error ek andsolve (3.8) on a coarser grid. Set dk−1 := rk−1,kdk and define ek−1 as the solution of

Lk−1ek−1 = dk−1 . (3.9)

An approximate solution of (3.8) is given by pk,k−1ek−1 and is used to correct to approx-imation uk

uk 7→ uk − pk,k−1ek−1.

A twogrid algorithm is the combination of such a coarse grid correction and the smoothingiteration. The twogrid iteration matrix Tk with ν1 pre-smoothing and ν2 post-smoothingsteps is given by

Tk(ν1, ν2) := LkKν2k

(L−1k − pk,k−1L

−1k−1rk−1,k

)LkK

ν1k . (3.10)

Kνk is the iteration matrix of the smoothing procedure K

(ν)k . We restrict to ν = ν1 pre-

smoothing steps and ν2 = 0. This restriction has no influence on the convergence analysis,because the spectral radius (Tk(ν1, ν2)) depends on ν := ν1 + ν2 only. In particular,(Tk(ν1, ν2)) = (Tk(ν)) holds, where Tk(ν) := Tk(ν, 0).

3.3.2 Abstract Multigrid Algorithm

The multigrid method is the recursive application of the twogrid algorithm to the defectequation (3.9) on coarser levels and can be written in the following form (cf. [Hac93,Section 2.5]):

procedure MG(uk, fk, ν);if k = 0 then u0 := L−1

0 f0 else

begin for i := 1 to ν do uk := Kk (uk, fk);dk−1 := rk−1,k (Lkuk − fk);

e(0)k−1 := 0;

for i := 1 to γ do e(i)k−1 := MG(e

(i−1)k−1 ,dk−1, ν);

uk := uk − pk,k−1e(γ)k−1;

end;

end.

Appropriate values for the iteration number γ are γ = 1 or γ = 2. For γ = 1 the multigridalgorithm MG can be written equivalently as

procedure MG(uk, fk, ν);begin for i := k step −1 to 1 do ui := Kν

i (ui, fi);fi−1 := ri−1,i (Liui − fi);end;

u0 := L−10 f0;

for i := 1 step 1 to k do ui := ui − pi,i−1ui−1;

end;

end.

(3.11)

3.4 Convergence Analysis 23

The sequence of operations during one step of this multigrid iteration with γ = 1 isdepicted in Figure 3.1 for the example k = 4. The stair symbolises the scale of levels. Forγ = 2 the multigrid algorithm is not in such a simple form as 3.11. Figure 3.2 shows thedetails of one multigrid step at level 4 involving 2 iterations at level 3, 4 iterations at level2 and 8 iterations at level 1. Due to the form of Figures 3.1 and 3.2 the iteration withγ = 1 is called V-cycle, and W-cycle for γ = 2.

S R

S R

SR

SR

E

P

P

P

P

k = 0

k = 1

k = 2

k = 3

k = 4

Figure 3.1: One multigrid iteration forγ = 1 and k = 4. S indicates thesmoothing step, R is the restriction ofthe defect, E is the exact solving at thelevel 0 and P stands for the correctionui 7→ ui − pi,i−1ui−1.

Figure 3.2: One multigrid iteration for γ = 2 and k = 4.

The efficiency of the multigrid algorithm depends on the convergence rate compared withthe computational work per iteration. The general convergence analysis in the frameworkof geometric multigrid methods is presented in the following section. In [Sau02], it wasshown that the computational amount of work for the composite finite element discreti-sation and the solution process via multigrid is proportional to the number of fine gridelements ♯Gkmax , i.e., the algorithm has quasi-optimal complexity.

3.4 Convergence Analysis

The presented multigrid algorithm fits in the framework of geometric multigrid methods(c.f. [Hac85]) and the convergence proof is based on that theory. However, one assumptionwhich is frequently used in [Hac85] is violated: The composite finite element spaces aredefined on grids which cover the domain Ω but do not necessarily resolve its geometry.The possible small intersection of a mesh cell with the domain leads to a non-standardscaling in the corresponding stiffness matrix entry and the convergence analysis has to beextended to cover such situations.

We will analyse the effect that, on coarse levels, large triangles might have an arbitrarysmall overlap with the domain. In order to adapt the classical multigrid convergencetheory we repeat the main results of this approach.

The multigrid convergence results from the convergence of the twogrid iteration. Usually,the convergence rate of the multigrid algorithm is only slightly worse than the twogridconvergence rate.


In the following subsections the general convergence analysis for the twogrid iteration andthe multigrid algorithm are explained in the framework of geometric multigrid methods.The content of these subsections is based on [Hac85, Chapters 6 and 7].

3.4.1 Twogrid Convergence

The twogrid iteration is a linear iteration having a representation

u(i+1)k = Tku

(i)k + Nkfk ,

where the iteration matrix Tk depends on the number ν of smoothing steps, Tk = Tk(ν).The twogrid iteration converges if and only if the spectral radius (Tk(ν)) < 1.Let Uk be the vector space of the grid functions uk endowed with the norm ‖ · ‖U . Inaddition, we introduce the vector space Fk consisting of the grid functions fk equippedwith the norm ‖·‖F . The standard case will be Uk = Fk and ‖·‖U = ‖·‖F = ‖·‖0,k, whichis the Euclidean norm. The induced matrix norms ‖ · ‖U←U , ‖ · ‖F←U etc. are defined by

‖A‖U←U := supv∈Uk\0

‖Av‖U‖v‖U

, ‖A‖F←U := supv∈Uk\0

‖Av‖F‖v‖U

. (3.12)

The estimate of ‖Tk‖U←U is often based on a suitable factorisation of Tk. The iterationmatrix Tk is split into the factors

Tk(ν) =(L−1k − pk,k−1L

−1k−1rk−1,k

)(LkK

νk) (3.13)

and this leads to

‖Tk(ν)‖U←U ≤ ‖L−1k − pk,k−1L

−1k−1rk−1,k‖U←F ‖LkKν

k‖F←U . (3.14)

Usually, the right-hand side of this inequality (3.14) is larger than the left-hand side, hencethe norm-estimates are somewhat pessimistic. However, for an appropriate choice of theinner norm ‖ · ‖F the estimate becomes arbitrarily sharp, since

‖Tk(ν)‖U←U = inf‖·‖F

‖L−1k − pk,k−1L

−1k−1rk−1,k‖U←F ‖LkKν

k‖F←U (3.15)

holds for the infimum taken over all norms.The second factor ‖LkKν

k‖F←U describes the efficiency of the smoothing behaviour of

K(ν)k . The smaller the other factor ‖L−1

k − pk,k−1L−1k−1rk−1,k‖U←F , the better the coarse

grid solutions approximate uk. Hence, the two essential parts of the twogrid iteration,namely the smoothing step and the coarse grid correction, can be analysed separately.

Definition 3.7 Let ‖ · ‖U and ‖ · ‖F be given. The procedure K(ν)k is has the smoothing

property if for each 0 < β < 1 there exist numbers 0 < ν < ν and 0 < ω < ω < 1independent of the iteration level k and hk and a number α such that

‖LkKνk‖F←U ≤ βh−αk ∀ν ∈ [ν, ν], ω ∈ [ω, ω] .

Definition 3.8 The approximation property holds if there is some constant Ca such that


−1k−1rk−1,k‖U←F ≤ Cah

αk ∀k ≥ 1

with α from Definition 3.7.

3.4 Convergence Analysis 25

The approximation property and the smoothing property are sufficient conditions for thetwogrid convergence.

Theorem 3.9 Suppose the smoothing property 3.7 and the approximation property givenin Definition 3.8. Let 0 < β < 1 be a fixed number.

1. ‖Tk(ν)‖U←U ≤ Caβ ∀ν ∈ [ν, ν], ω ∈ [ω, ω] .

2. Since the solution of Lkuk = fk is a fixed point of the smoothing procedure K(ν)k one

obtains convergence

uik → uk = L−1k fk as i→ ∞ .

The first assertion is a combination of smoothing and approximation property (see (3.14).For a proof of the second assertion we refer to [Hac85, Theorem 6.1.7].

3.4.2 Multigrid Convergence

In this thesis we restrict to the analysis of the multigrid convergence of a W-cycle (c.f.Figure 3.2). In [Hac85, Chapter 7] it is shown, that the sufficient conditions for thetwogrid method from Theorem 3.9 almost imply the multigrid convergence. In additionto the assumptions of this theorem one needs

‖K(ν)k ‖U←U ≤ CS ∀k ≥ 1, 0 < ν < ν(hk) . (3.16)

Further, we require the inequalities

Cp‖uk−1‖U ≤ ‖pk,k−1uk−1‖U ≤ Cp‖uk−1‖U (3.17)

with Cp and Cp independent of k. The assertions about the twogrid convergence given inTheorem 3.9 can also be established for the multigrid case with these assumptions. LetMGk denote the iteration matrix of the W-cycle multigrid algorithm.

Theorem 3.10 Suppose γ ≥ 2, (3.16), (3.17), the smoothing property 3.7 and the ap-proximation property (c.f. Definition 3.8). Let 0 < ζ < 1 be a fixed number.

1. There exist 0 < ω < ω < 1, and 1 < ν < ν independent of k such that

‖MGk‖0,1←0,1 ≤ ζ ∀ q ∈ ]0, q] , ν ∈ [ν, ν] , ω ∈ [ω, ω] .

2. Since the solution of Lkuk = fk is a fixed point of the smoothing procedure K(ν)k one

obtains convergence

uik → uk = L−1k fk as i→ ∞ .

For the proof we refer to [Hac85, Theorem 7.1.2]. As mentioned at the beginning of thissection the application of the classical multigrid convergence theory to composite finiteelements is by no means trivial since the composite finite element spaces are defined ongrids which overlap the domain but do not necessarily resolve the geometry of Ω. Theconvergence analysis for two model problems, which allow the study of the characteristicdifficulties systematically will be worked out in Chapter 5 and Chapter 6.


4 Numerical Experiments

We presented in Chapter 3 a multigrid solver for systems of linear equations arising bydiscretising elliptic boundary value problems on complicated domains. We restrict toproblems with smooth coefficients so that a Poisson-type problem on a complicated domainis an adequate model problem (c.f. (1.1)). For the solution of problems with jumpingcoefficients we refer to [War03] and [SW04]. We assumed that the discretisation of theproblem, i.e., the finite element mesh and the system of equations is given without havinga hierarchy of coarse scale discretisations at hand. For the construction of coarse scalediscretisations composite finite elements were used (see Section 2.2.2). The composite finiteelement spaces are defined on grids which overlap the domain but do not necessarily resolveit. The multigrid method for composite finite elements is quite simple. No modificationor complicated smoothing or coarsening strategy has to be employed. On the other hand,as a consequence of the simplicity of the method, it turns out, the convergence analysis isnon-trivial and the classical theory of geometric multigrid algorithms cannot be appliedin a straightforward fashion. The reason is that some degrees of freedom lie outside thedomain and the intersection of the support of the corresponding basis functions with thedomain is small.The scope of this work is to prove that the convergence behaviour of this multigrid algo-rithm is independent of the number and sizes of the geometrical details of the domain atthe boundary. First, we will derive some numerical results to work out the main criteria.In the second part of this chapter we formulate specific model problems which allow thesystematic study of the small overlaps on the convergence rates.

4.1 Numerical Example

In [NS05] the results of numerical experiments are reported and we include them hereas a motivation. As a practical application in environmental modelling the convergencebehaviour of the multigrid method based on composite finite elements for a Neumannproblem on the complicated domain of the Baltic sea (cf. Figure 4.1) has been studied.The test problem was

−∆u+ u = 1 in Ω

∂u/∂n = 0 on ∂Ω .

The V-cycle multigrid algorithm was applied with two symmetric Gauß-Seidel smoothingsteps. The stopping criterion for the algorithm was ‖Lkmaxu

(i) − fkmax‖ ≤ 10−8.The results in Table 4.1 show, that the number of multigrid iterations is bounded andsmall independent of the number of unknowns.In Figure 4.2 the grid sequence is depicted which shows that the coarsest grid has onlynine degrees of freedom. The overlaps of triangles with the domain Ω are of rather general

27

28 4 Numerical Experiments

Figure 4.1: Ω is the two-dimensional surface of the Baltic sea.

Refinement

level2 3 4 5 6 7 8 9 10 11 12 13 14 15

♯ mg

iterations6 7 12 16 13 20 22 17 18 17 16 15 13 13

Table 4.1: Convergence rates for applying the multigrid algorithm to the test problem onthe Baltic sea.

shape. Since the grid Gk is an overlapping triangulation of the domain, the intersectionof triangles at the boundary with the domain might be arbitrarily small and of generalshape. In Figure 4.3 a characteristic situation is depicted.

Elements τ of the auxiliary grid Gk might have a very small overlap |τ∩Ω| with the domainΩ. Hence, the degrees of freedom lying (essentially) outside of the domain lead to a verydifferent scaling in the corresponding matrix entries compared to the interior unknowns.In standard cases, all entries of the system matrix Lk are O(1). Robust convergence canbe proved with standard convergence theory (c.f. Section 3.4) as long as

⋃

x∈Θk

((suppϕx,k) ∩ Ω) = Gk .

The case that (suppϕx,k)∩Ω is degenerated, i.e., much smaller than h2k, for some x ∈ Θk

is illustrated in Figure 4.3. The intersection (suppϕx,k) ∩ Ω for the vertex x of triangle τis of order ε2 with 0 < ε < αh with 0 < α < 1 fixed. The gradient of the shape functionon τ is of order h−1. Therefore the corresponding matrix entry (L)xx is of order (ε/h)2.Compared to the matrix entry (L)yy corresponds to the node y ∈ Θ which is of order 1,we have a scaling factor (ε/h)2. This declined the condition number of the system matrixL. We will prove for some appropriate model problems that the multigrid convergence isnot affected by such scaling effects.

4.2 Model Problems 29

Figure 4.2: Overlapping grids Gk on discretisation level k = 1, k = 2, k = 3, k = 4 andk = 10 (from left to right).

x y

z

τ

Figure 4.3: The overlaps of triangles τ ∈ G with the domain Ω are of rather general shape.

4.2 Model Problems

Motivated by Figure 4.3 we have designed model problems which exhibit the followingcharacteristic situations:

1. Nodes of type “y”: The intersection (suppϕy,k) ∩ Ω has measure h2k but is not the

union of triangles τ in Gk.


2. Nodes of type “z”: The intersection (suppϕz,k)∩Ω has measure εh2k where ε denotes

the depth of penetration of the relevant triangle, see Figure 4.4.

3. Nodes of type “x”: The intersection (suppϕx,k) ∩ Ω has measure ε2.

z

τε

Figure 4.4: The depth ε of penetration of triangle τ .

4.2.1 2D Model Problem

Let Ω = (0, 1 + ε) × (0, 1) ⊂ R2 with 0 < ε≪ 1. Consider the Poisson equation

−∆u = f in Ω

with homogeneous Neumann boundary conditions ∂u/∂n = 0 on ΓN := 1 + ε × (0, 1)and Dirichlet boundary conditions u = 0 on ΓD := Γ \ ΓN.

Problem 4.1 Let f ∈ L2(Ω) be given. We seek a function u ∈ H1D(Ω) so that

a(u, v) =

∫

Ω

fvdx , ∀ v ∈ H1D(Ω) .

From Proposition 1.4 the existence and uniqueness of a solution of this problem follow.

For the construction of a hierarchy of overlapping grids we start with an initial grid G0 =τ1, τ2, . . . , τ12 (see Figure 4.5). By connecting midpoints of the edges of the elementsof coarser triangulations we generate the next finer grid. Triangles having no intersectionwith the domain are removed from these grids. Then, (Gk)0≤k≤kmax

is a sequence ofoverlapping grids. The grids Gk are nested for 0 ≤ k ≤ kmax but, in general, they donot resolve the Neumann boundary ΓN . The domain covered by the grid Gk is denotedby Ωk and Θk denotes the set of vertices of triangles in Gk. The number of unknowns isnk = ♯Θk = 22(k+1) − 1 and the step size is hk = 2−k−1. The definition of the auxiliarygrids Gk implies for all 0 ≤ k ≤ kmax

Ω ⊂ Ωk+1 ⊂ Ωk and Θk+1 ⊂ Ωk .


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ΓD ΓN

τ1

τ2

τ3

τ4

τ5 τ9

τ10

τ11

(0, 0)

(0, 1)

(1, 0) (1 + ε, 0)

1 2 3

Figure 4.5: Initial grid on discretisation level k = 0.

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Ω

ΓD

Γk

(0, 0)

(0, 1)

(1, 0)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Figure 4.6: Overlapping grid on discretisation level k = 1.

The construction of Problem 4.1 implies that triangles at the right boundary of Ωk e.g.Γk := 1+hk× (0, 1) have only a small overlap with the domain Ω. On the coarsest gridG0 the triangles τ9 and τ11 have intersection |τ ∩Ω| with the domain of order ε2 (see Figure4.5). The measure of the overlap of the triangles τ10 and τ12 with the domain Ω is of orderεhk. Therefore, the corresponding system matrix entry for the degree of freedom 3 is oforder ε/hk, and hence degenerated. The next finer grid leads to the same scaling effect inthe system matrix L1. The corresponding matrix entries for the degrees of freedom 13, 14and 15 are of order ε/hk, as all other matrix entries are O(1) (see Figure 4.6). Hence, thescaling factor in the system matrix Lk is ε/hk for all 0 ≤ k ≤ kmax. Define

qk :=ε

hk. (4.1)

The goal is to analyse the influence of the scaling effect in the system matrix on theconvergence rate of the multigrid algorithm. Therefore, we consider qk as a parameterand prove the robust multigrid convergence independent of qk and the maximal numberof discretisation levels kmax.

The composite finite element space SCFEk (Ω) is given by

SCFEk (Ω) = S(Ωk)|Ω = u|Ω : u ∈ S(Ωk)


(c.f. Subsection 2.2.2). In the sequel we write Sk(Ω) instead of SCFEk (Ω). The finite

element discretisation of Problem 4.1 is given by seeking uk ∈ Sk(Ω) such that

a(uk, vk) =

∫

Ω

fvk dx ∀ vk ∈ Sk(Ω) .

A basis of Sk(Ω) is ϕk,i(x)|Ωx∈ΘkIn Figures 4.7 and 4.8 the basis functions on the

Neumann part of the boundary ΓN are depicted.

Figure 4.7: The basis functions of the composite finite elements at the Neumann boundaryon a triangle τ with overlap |τ ∩ Ω| = O(ε2).

Figure 4.8: The basis functions of the composite finite elements at the Neumann boundaryon a triangle τ with overlap |τ ∩ Ω| = O(εhk).

The ansatz

uk = Pkuk :=

nk∑

i=1

(uk)iϕk,i|Ω

leads to the system of linear equations

Lkuk = fk ,

with

(Lk)i,j := a (ϕk,j|Ω, ϕk,i|Ω) , (fk)i :=

∫

Ω

fϕk,i|Ωdx ∀ 1 ≤ i, j ≤ nk .


ε‖Tk‖0←0 0.6 · 10−2 0.6 · 10−3 0.6 · 10−4 0.6 · 10−5 0.6 · 10−6

2 0.43 0.45 0.45 0.45 0.453 0.42 0.45 0.45 0.45 0.45

k4 0.40 0.44 0.45 0.45 0.455 0.37

Table 4.2: ‖Tk‖0←0 with fixed ω = 0.5 and ν = 4.

ε‖V‖0←0 0.6 · 10−2 0.6 · 10−3 0.6 · 10−4 0.6 · 10−5 0.6 · 10−6

2 20.55 208.07 2.08 · 103 2.08 · 104 2.08 · 105

3 10.11 103.90 1.04 · 103 1.04 · 104 1.04 · 105

k4 4.88 51.81 520.57 5.21 · 103 5.21 · 104

5 2.27 25.76 260.15 2.60 · 103 2.60 · 104

Table 4.3: ‖V‖0←0 with fixed ω = 0.5 and ν = 4.

First, we analyse the convergence of the twogrid iteration with respect to the levels k,k − 1. As the intergrid transfer operator we use the canonical finite element prolongationpk,k−1 : R

nk−1 → Rnk as defined in (3.1) and as the restriction rk−1,k its transposed. The

twogrid iteration matrix with ν (pre-)smoothing steps is given by

Tk = Tk(ν) :=(

L−1k − pk,k−1L

−1k−1p

⊺

k,k−1

)

LkKνk ,

with Kk the iteration matrix of the damped Jacobi iteration.Let U be a Hilbert space with norm ‖ ·‖U . We prove the twogrid convergence with respectto the Euclidean norm ‖ · ‖0←0. The convergence theory of geometric multigrid is basedon the splitting

‖Tk‖0←0 ≤ ‖L−1k − pk,k−1L

−1k−1p

⊺

k,k−1‖0←U· ‖LkKν

k‖U←0 . (4.2)

We define

V := L−1k − pk,k−1L

−1k−1p

⊺

k,k−1 . (4.3)

The most simple choice for U , namely, Rnk with the Euclidean norm ‖ · ‖U = ‖ · ‖0 is not

possible. The numerical results in Tables 4.4 and 4.3 show, that ‖LkKνk‖0←0 is bounded

by a constant but the factor ‖L−1k − pk,k−1L

−1k−1p

⊺

k,k−1‖0←0diverges as ε→ 0.

Hence, we shall introduce a weighted norm to specify the inner norm ‖ · ‖U in (4.2). Wedefine on R

nk

‖|u‖|0 := ‖D−1k u‖0 , (4.4)

where Dk is the diagonal part of Lk. The norm ‖| · ‖|0 corresponds to a weighted Euclideannorm. Applying the definition (4.4) yields

‖Tk‖0←0 =∥∥∥

(


−1k−1p

⊺

k,k−1

)

DkD−1k LkK

νk

∥∥∥

0←0

≤∥∥∥

(


−1k−1p

⊺

k,k−1

)

Dk

∥∥∥

0←0·∥∥D−1

k LkKνk

∥∥

0←0. (4.5)


ε‖LkKνk‖0←0 0.6 · 10−2 0.6 · 10−3 0.6 · 10−4 0.6 · 10−5 0.6 · 10−6

2 0.65 0.65 0.65 0.65 0.653 0.65 0.65 0.65 0.65 0.65

k4 0.66 0.66 0.66 0.66 0.665 0.66

Table 4.4: ‖LkKνk‖0←0 with fixed ω = 0.5 and ν = 4.

ε‖VDk‖0←0 0.6 · 10−2 0.6 · 10−3 0.6 · 10−4 0.6 · 10−5 0.6 · 10−6

2 2.69 2.72 2.72 2.72 2.723 2.91 2.92 2.92 2.92 2.92

k4 3.13 3.14 3.14 3.14 3.145 3.27

Table 4.5: ‖VDk‖0←0 with fixed ω = 0.5 and ν = 4.

ε‖D−1k LkK

νk‖0←0 0.6 · 10−2 0.6 · 10−3 0.6 · 10−4 0.6 · 10−5 0.6 · 10−6

2 0.22 0.22 0.22 0.22 0.223 0.21 0.23 0.23 0.23 0.23

k4 0.20 0.22 0.23 0.23 0.235 0.18 0.22 0.23 0.23 0.23

Table 4.6: ‖D−1k LkK

νk‖0←0 with fixed ω = 0.5 and ν = 4.

In Chapter 6 we prove for this two-dimensional model problem the robust convergence ofthe multigrid algorithm with respect to a small overlap of triangles with the domain atthe Neumann boundary. The application of the classical multigrid convergence theory isnot possible in a straightforward manner. To get an idea, how to adapt the classical con-vergence theory to our specific situation we first analyse a one-dimensional model problemwhich can be viewed as a cross-section of the grid and domain of the two-dimensionalmodel problem.

4.2.2 1D Model Problem

Let Ω = (0, 1) ⊂ R and set H1D(Ω) := u ∈ H1(Ω) : u(1) = 0. Consider

Problem 4.2 Find, for given f ∈ L2 (Ω), a function u ∈ H1D(Ω) so that

a(u, v) =

∫

Ω

fv dx ∀ v ∈ H1D with a(u, v) :=

∫

Ω

u′v′dx .


r r r r r r

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x1,0 0 x1,1 = ε x1,2 x1,3 x1,4 x1,5 = 1

Figure 4.9: Overlapping grid G1 is generated from the initial grid G0.

The construction of a hierarchy of triangulations starts with the initial grid G0 = τ0,i1≤i≤3

with τ0,i := (x0,i−1, x0,i). We assume that the triangle τ0,1 has only a small overlap withthe domain at the Neumann boundary

|τ0,1 ∩ (0, 1)| =: ε < αh0,1 with 0 < α < 1 fixed ,

where h0,1 is the length of τ0,1. In addition we assume x0,3 = 1. The next finer tri-angulation G1 is generated from G0 by introducing midpoints to the intervals τ0,i for all1 ≤ i ≤ 3. Triangles having no intersection with domain are removed from the grid. InFigure 4.9 the grid G1 is depicted. Further refinements lead to a sequence of overlappinggrids (Gk)0≤k≤kmax

. The grid points are numbered from left to right and we assume

xk,0 < 0 < xk,1 < xk,2 < · · · < xk,nk= 1 .

The intervals are given by τk,i := (xk,i−1, xk,i) with lengths hk,i := xk,i − xk,i−1. Themaximal step size is hk := max1≤i≤nk

hk,i. Direct consequences of the construction of thegrid hierarchy are that xk,1 = ε for all 0 ≤ k ≤ kmax and that all grids Gk resolve theDirichlet boundary, i.e., xk,nk

= 1 for all 0 ≤ k ≤ kmax. More precisely, we assume

0 < ε < αhk+1,1 < 1 ∀ 0 ≤ k ≤ kmax , (4.6)

with 0 < α < 1 fixed, and therefore,

0 < ε < αhkmax,1 < 1 .

Due to the Dirichlet boundary conditions, the degrees of freedom are associated with theset of grid points Θk := xk,i : 0 ≤ i ≤ nk − 1. The domain which is covered by the gridGk is denoted Ωk.We shall regard the (small) overlap ε of the first interval τk,1 with the domain as a param-eter and investigate the behaviour of the multigrid convergence as ε tends to 0.The composite finite element space SCFE

k (Ω) is simply the restriction of the standard finiteelement space on the overlapping grid Gk to (0, 1)

SCFEk (Ω) := Sk(Ωk)|Ω := u|Ω : u ∈ Sk(Ωk) .

The finite element discretisation is given by seeking uk ∈ SCFEk (Ω) such that

a(uk, vk) =

∫

Ω

fvk dx ∀ vk ∈ SCFEk (Ω) .

As a basis of SCFEk (Ω) we choose the usual nodal basis ϕk,ink

i=1, defined in (1.16), on theoverlapping grid Gk, where the basis functions ϕk,1 and ϕk,2 are restricted to the domain


r r r r r r

.

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..

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xk,0 0 xk,1 = ε xk,2 xk,nk−2 xk,nk−1 1 = xk,nk

ϕk,1

ϕk,2 ϕk,n

Figure 4.10: The basis functions of the composite finite elements at the boundary. Theunderlying grid Gk = τk,1, . . . , τk,nk

does not resolve the Neumann boundary ΓN.

Ω (see Figure 4.10). Applying the basis representation of uk and testing with the basisfunction leads to the system of linear equations

Lkuk = fk ,

with

(Lk)i,j := a (ϕk,i, ϕk,j) , (fk)i :=

∫

Ω

fϕk,idx ∀ 0 ≤ i, j ≤ nk − 1 .

The twogrid iteration matrix with ν smoothing steps is given by

Tk(ν) :=(L−1k − pk,k−1L

−1k−1rk−1,k

)LkK

νk .

ε‖Tk‖0←0 0.1 0.01 0.001 0.0001

1 2.15 7.04 22.36 70.712 1.07 3.52 11.18 35.35

ν3 0.42 1.39 4.42 13.974 0.22 0.70 2.20 6.99

Table 4.7: ‖Tk‖0←0 with fixed ω = 0.5 and k = 3.

ε‖Tk‖0,1←0,1 0.1 0.01 0.001 0.0001

1 0.48 0.48 0.48 0.482 0.26 0.24 0.24 0.24

ν3 0.15 0.14 0.14 0.144 0.11 0.11 0.11 0.11

Table 4.8: ‖Tk‖0,1←0,1 with fixed ω = 0.5 and k = 3.

In the following Chapter 5 it is proved for this one-dimensional model problem that themultigrid convergence is robust with respect to a small overlap of elements with the domain


by using the weighted Euclidean norm ‖ · ‖0,1 (c.f. Definition 5.1). Although, the modelproblem has full regularity, i.e., ‖u‖H2(Ω) ≤ C‖f‖L2(Ω), the use of the discrete Euclideannorm for the smoothing and the approximation property is not appropriate: Numericalexperiments (c.f. Table 4.7) show that ‖Tk‖0←0 diverges as ε→ 0 and we have to employnorms with appropriate weights at the Neumann boundary. With respect to this weightednorm ‖ · ‖0,1 the twogrid method converges robust (see Table 4.7).


5 Convergence Analysis for the

1D Model Problem

First, we will analyse the convergence of a twogrid algorithm with respect to the levels k,k − 1. As the intergrid transfer operator we use the canonical finite element prolongation(cf. Definition 3.1) pk,k−1 : R

Θk−1 → RΘk and, as the restriction rk−1,k, its transposed.

The twogrid iteration matrix with ν pre-smoothing steps is given by

Tk(ν) :=(L−1k − pk,k−1L

−1k−1rk−1,k

)LkK

νk . (5.1)

Our aim is to prove the twogrid convergence by investigating the smoothing and theapproximation property with respect to appropriate norms (c.f. Section 3.4.1). Numericalresults (c.f. Section 4.2.2) show, that the discrete Euclidean norm is not appropriatebecause ‖Tk,k−1‖0←0 diverges as ε, i.e. the intersection of the interval τk,1 with thedomain Ω, tends to zero, while the spectral radius is bounded away from 1 independentof ε. Below, we will prove the robust convergence as ε→ 0 in a weighted norm, where theleft-most unknown, lying essentially outside Ω, is damped.

Definition 5.1 For s ∈ R, let

Ns,k := hk diag [q3−sk , 1, 1, . . . , 1] ,

with qk := ε/hk,1. We define a scale of mesh-dependent scalar products and norms on Rnk

in the following way

〈u,v〉0,s = u⊺Ns,kv and ‖u‖0,s := 〈u,u〉1/20,s ∀u,v ∈ RΘk .

The indices 0, s characterise the correspondence of the norm ‖ · ‖0,s to a weighted Euclideannorm.

Remark 5.2 It can be shown that the norms ‖Pk (·)‖L2(Ω) and ‖·‖0,0 are equivalent, wherePk is defined in (2.8) with respect to the discretisation level k.

In the following two subsections we will prove the smoothing and the approximationproperty for the twogrid iteration with respect to this weighted Euclidean norm.

Lemma 5.3 For all β ∈ (0, 1) there exist 0 < ω < ω < 1, 1 < ν < ν, and q > 0independent of k such that

‖LkKνk‖0,1←0,1 ≤ βh−1

k ∀ q ∈ ]0, q] , ν ∈ [ν, ν] , ω ∈ [ω, ω] . (5.2)

39

40 5 Convergence Analysis for the 1D Model Problem

Lemma 5.4 The approximation property holds with respect to the ‖ · ‖0,1-norm


−1k−1p

Tk,k−1‖0,1←0,1

≤ Cahk ,

where the constant Ca is independent of k and qk.

Adapting the general convergence analysis for the twogrid algorithm explained in Section3.4.1 these two properties lead to the convergence of the twogrid method with respect tothe ‖·‖0,1-norm uniformly with respect to the relative overlap qk ∈ ]0, q] with q = O(1).The proofs of the both convergence properties are based on a splitting of the systemmatrix into a part, which describes the standard finite element discretisation of a reducedproblem, and the discretisation of the near boundary part. Explicit calculations yield thatthe system matrix Lk has the following representation

Lk =

0 . . . 0... Ak

0

+

qkhk,1

1 −1−1 1

0

0 0

, (5.3)

where Ak denotes the finite element discretisation of a reduced Poisson problem

−u′′ = f in (ε, 1) ,

u′(ε) = u(1) = 0(5.4)

on the sub-mesh Gk := τ ∈ Gk : τ ⊂ (ε, 1).

5.1 Proof of the Smoothing Property

To simplify the notation we skip the index k in this section. From the definition of thenorm ‖ · ‖0,s it follows

‖LKν‖0,1←0,1 = ‖N1/21 LKνN

−1/21 ‖0←0 .

The ansatz is to use an appropriate multiplicative splitting of this matrix

N1/21 LKνN

−1/21 = N

1/21 L

(I − ωD−1L

)νN−1/21

= N1/21 D1/2X (I − ωX)ν D1/2N

−1/21

= N1/21 D1/2N

−1/41 W (I− ωW)ν N

1/41 D1/2N

−1/21

with X := D−1/2L D−1/2 and W := N1/41 X N

−1/41 . Therefore,

‖LKν‖0,1←0,1 ≤ ‖N1/21 D1/2N

−1/41 ‖0←0 · ‖W (I − ωW)ν ‖0←0

· ‖N1/41 D1/2N

−1/21 ‖0←0 .

Explicit calculations for the first and the last factor yield

‖LKν‖0,1←0,1 ≤ C0h−1‖W (I − ωW)ν ‖0←0 , (5.5)

while C0 is independent of k and q.

5.1 Proof of the Smoothing Property 41

It remains to estimate ‖W (I − ωW)ν ‖0←0. By using the partitioning of the systemmatrix (5.3) we have

W =

0 . . . 0... D

−1/2A AD

−1/2A

0

+

1 W12 0W21 0 W23

0 W32 00

0 0

3

n− 3

︸︷︷︸

3︸︷︷︸

n−3

with DA := diag(A). We define the normalised stiffness matrix

A := D−1/2A AD

−1/2A ∈ R

(n−1)×(n−1) . (5.6)

The matrix entries W12, W23 and W32 are of order O(q) and |W21| = O(1). Let ei denotethe i-th canonical unit vector in R

n−1. We apply the splitting

W =

[1 0

W21e1 A

]

+

1 W12 00 0 W23

0 W32 00

0 0

=: WI + WII (5.7)

This leads to

W (I − ωW)ν = WI (I − ωWI − ωWII)ν + WII (I − ωW)ν .

The first term on the right-hand side above can be rewritten as a sum of terms of the form

WI

t∏

i=1

(I − ωWI)αi (−ωWII)

βi (5.8)

with αi, βi ∈ N0 and |α| + |β| = ν. Here, |µ| :=∑t

i=1 µi. To avoid the identity matrix inbetween of the factors in (5.8) we may assume, w.l.o.g.,

αi+1, βi > 0 ∀ 1 ≤ i ≤ t− 1 .

There exists only one summand with |β| = 0, namely

WI (I − ωWI)ν .

This leads to the estimate

‖W (I − ωW)ν‖0←0 ≤ ‖WI (I− ωWI)ν‖0←0

+ν−1∑

j=0

(ν

j

)

‖WI(I − ωWI)‖j0←0‖ωWII‖ν−j0←0

+ ‖WII‖0←0‖I − ωW‖ν0←0

(5.9)

and the single factors and sums on the right-hand side in (5.9) will be considered below.


We begin with the estimate of WI (I − ωWI)j and (I − ωWI)

j . Define

B := (1 − ω)−1(

I − ωA)

,

b(j) := W21e1 −ωW21

1 − ωA

j−1∑

m=0

Bme1 and

c(j) := − ω

1 − ωW21

j−1∑

i=0

Bie1 .

(5.10)

Explicit calculations and Lemma A.2 yield

(I − ωWI)j = (1 − ω)j

[1 0

c(j) Bj

]

and

WI (I − ωWI)j = (1 − ω)j

[1 0

b(j) ABj

]

.

(5.11)

A is symmetric, because it is the associate system matrix of the standard finite elementdiscretisation of the reduced Poisson problem (5.4). Since therefore the matrix A issymmetric its spectral radius equals ‖A‖0←0.

Lemma 5.5 A is weakly diagonal dominant and for the spectral radius of the product Aholds

(

A)

≤ 2 .

Proof. From the definition of the bilinear form a follows for

i = 1 :n∑

j=1|(A)ij | = a(ϕ1, ϕ1) + |a(ϕ1, ϕ2)| = 3 < 2(A)11 ,

i = n :n∑

j=1|(A)ij | = a(ϕn, ϕn) + |a(ϕn−1, ϕn)| = 3 < 2(A)nn and

2 ≤ i ≤ n− 1 :n∑

j=1|(A)ij | = a(ϕi, ϕi) + |a(ϕi−1, ϕi)| + |a(ϕi, ϕi+1)| = 4 = 2(A)ii .

This means A is weakly diagonal dominant. Now, for the spectral radius of A, we obtain

(

A)

= ((

A1/2D−1/2A

)⊺

A1/2D−1/2A

)

=∥∥∥A1/2D

−1/2A

∥∥∥

2

0←0

= supv∈Rn\0

〈A1/2D−1/2A v,A1/2D

−1/2A v〉

〈v,v〉 = supu∈Rn\0

〈A1/2u,A1/2v〉〈D1/2

A v,D1/2A v〉

= supu∈Rn\0

〈Au,v〉〈DAv,v〉 .

With the inequality st ≤ 1/2(s2 + t2) for s, t ∈ R, the symmetry and the weakly diagonal


dominance of A follows

〈Au,u〉 =

n∑

i=1

(Au)i (u)i =

n∑

i=1

n∑

j=1

(A)ij(u)jui ≤n∑

i=1

n∑

j=1

|(A)ij |(u)2j + (u)2i

2

≤ 1

2

n∑

j=1

(u)2j

n∑

i=1

|(A)ij | +1

2

n∑

i=1

(u)2i

n∑

j=1

|(A)ij |

≤n∑

j=1

(u)2j |(A)jj | +n∑

i=1

(u)2i |(A)ii|

= 2

n∑

i=1

(u)2i |(A)ii| = 2

n∑

i=1

(DA)ii(u)2i = 2 〈DAu,u〉 .

Combining these results yields the assertion.

The eigenvectors of B coincide with those of A and the eigenvalues of B are of the form

(1 − ωλ)/(1 − ω) with λ ∈ spec(

A)

. Thus, the Euclidean norm of B is bounded by

(1 − ω)−1 for all 0 < ω ≤ ω ≤ 1/2.

From [Hac85, Lemma 1.3.5 ], we conclude that for any 0 < ω ≤ ω and ω ∈ [ω, ω],

‖ABj‖0←0

=

∥∥∥∥A

1

(1 − ω)j(I − ωA)j

∥∥∥∥

0←0

≤ 1

(1 − ω)jmaxfj(λ) = λ(1 − ωλ)j : λ ∈ spec(A) .

The maximum above is bounded by maxfj(λ) : λ ∈ spec(A) ≤ (ω(1 + j))−1, i.e.,

‖ABj‖0←0 ≤ 1

ω (1 − ω)j (j + 1). (5.12)

Next, the Euclidean norms of the vectors b(j) and c(j) in (5.11) are estimated.

Lemma 5.6 For all ω ≤ 1/2 there exists a maximal number of smoothing steps ν > 1such that for all ω ≤ ω and j ≤ ν it holds

‖b(j)‖0 ≤ C11

ω3(j + 1)(1 − ω)jand ‖c(j)‖0 ≤ C2

1

(1 − ω)j, .

where C1 and C2 are constants independent of ω and j.

Proof. Let (µi)n−1i=1 denote the (orthonormal) eigenvectors of A and expand e1 according

to e1 =∑n−1

i=1 αiµi. Thus,

b(j) = W21e1 −ωW21

1 − ω

n−1∑

i=1

j−1∑

m=0

αiλi

(1 − ωλi1 − ω

)m

µi


and, for the Euclidean norm, one gets

‖b(j)‖20 ≤ 2

|W21|2‖e1‖20 +

∥∥∥∥∥

ωW21

1 − ω

n−1∑

i=1

j−1∑

m=0

αi(1 − ω)m

|fm(λi)|µi∥∥∥∥∥

2

0

= 2|W21|2

1 +

(ω

1 − ω

)2 n−1∑

i=1

α2i

(j−1∑

m=0

|fm(λi)|(1 − ω)m

)2

≤ 2|W21|2

1 +

(ω

1 − ω

)2 n−1∑

i=1

α2i

(j−1∑

m=0

1

ω (1 +m) (1 − ω)m

)2

.

For all 0 < ω ≤ ω ≤ 1/2 there exists a maximal number of smoothing steps ν > 1 suchthat for all m ≤ ν holds (1 + ω)m ≤ m+ 1. Let the minimal number of smoothing stepsν ≥ 1 be chosen such that ω(ν + 1) ≤ 1. Then, for all j ≥ ν it holds

ω(j + 1) ≤ (1 + ω)j .

Combining these two inequalities we obtain for all m, j ∈ [ν, ν] and ω ≤ ω

ω(j + 1)

m+ 1≤ (1 + ω)j−m .

Straightforward analysis yields

j−1∑

m=0

(1 − ω)j−mj + 1

m+ 1≤ 1

ω

j−1∑

m=0

(1 − ω)j−m (1 + ω)j−m =1

ω

j−1∑

m=0

(1 − ω2

)j−m

=1

ω

(j∑

i=0

(1 − ω2

)i − 1

)

=1 −

(1 − ω2

)j+1

ω3− 1

ω≤ 1 − ω2

ω3.

Using this estimate and∑α2i = 1, the norm of b(j) is bounded by

‖b(j)‖0 ≤ C11

ω3 (j + 1) (1 − ω)j,

where the constant C1 is independent of ω and j for all ω ≤ ω and j ∈ [ν, ν]. Next we willprove the second assertion. From the definition of c(j) in (5.10) and the estimate (5.12)one concludes

‖c(j)‖20 ≤ C2

2

(ω

1 − ω

)2(j−1∑

m=0

1

(1 − ω)m

)2

.

Applying a geometric sum argument we obtain

j−1∑

m=0

(1 − ω)j−m =

j∑

i=0

(1 − ω)i − 1 ≤ 1 − ω

ω.

The combination of these two estimates yields ‖c(j)‖0 ≤ C2(1 − ω)−j .

5.2 Proof of the Approximation Property 45

By using the results of Lemma 5.6 and equation (5.7) we derive∥∥∥WI (I− ωWI)

j∥∥∥

0←0≤ CI(1 − ω)j

(

1 +∥∥∥b(j)

∥∥∥

0+∥∥∥ABj

∥∥∥

0←0

)

≤ CI1

ω3 (j + 1),

∥∥∥(I− ωWI)

j∥∥∥

0←0≤ C3 = O(1) and

∥∥∥W

jII

∥∥∥

0←0≤ (CIIq)

j .

(5.13)

It remains to estimate the norm ‖I − ωW‖ν0←0 in (5.9). We obtain with the result ofLemma 5.5

‖W‖0←0 ≤ ‖WI‖0←0 + ‖WII‖0←0 ≤ C(

1 +∥∥∥A∥∥∥

0←0+ q)

≤ C(3 + q) .

Thus, the choice in (5.2)

ω := min

2

C(3 + q),1

2

(5.14)

leads to

‖I − ωW‖ν0←0 ≤ 1 .

Combining this estimate and (5.13) with (5.9) results in

‖W (I − ωW)ν‖0←0 ≤ Cs

(

1ω3(ν+1)

+ν−1∑

j=0

(νj

)(CIIωq)

ν−j + q

)

≤ Cs

(1

ω3(ν+1) + ((1 + CIIωq)ν − 1) + q

)

, (5.15)

where Cs is a constant independent of q and h.Next, we will determine the ranges of ω, ν and q such that every term in (5.15) is boundedby β/3 for any 0 < β < 1.

Fix 0 < ω < ω and 0 < β < 1. Put ν := 3C1

(ω2β

)−1 − 1 > 0 and fix O(1) = ν > ν.Then, a tedious analysis of the right hand side in (5.15) yields

‖W (I − ωW)ν‖0←0 ≤ β ∀ q ∈ ]0, q] , ν ∈ [ν, ν] , ω ∈ [ω, ω] ,

with

q = min

β

3C1,

1

CIIω

((

1 +β

3Cs

)1/ν

− 1

)

= O(1) .

By replacing β by β/C0 with C0 as in (5.5) yields the proof of the smoothing propertywith α = 1 (see Definition 3.7 and Lemma 5.3).

5.2 Proof of the Approximation Property

We again employ the block partitioning

Lk =

0 . . . 0... Ak

0

+

qkhk,1

1 −1−1 1

0

0 0

.


Let ek,i denote the ith canonical unit vector in Rnk−1. Explicit calculations leads to the

following representation of the inverse of the stiffness matrix

L−1k =

hk,1

qk+(A−1k

)

11

(A−1k ek,1

)⊺

A−1k ek,1 A−1

k

.

Next, we compute the coarse grid correction matrix Vk := L−1k − pk,k−1L

−1k−1p

⊺

k,k−1. Ac-cording to the partitioning of the system matrix (5.3) we introduce an associated parti-tioning of the prolongation operator:

pk,k−1 =

12

12 0 . . . 0

0 pk,k−1

,

with the (canonical) prolongation pk,k−1 corresponding to the reduced grids Gk−1, Gk.Explicit calculations result in

Vk=

hk,1

qk+(A−1k

)

11

(A−1k ek,1

)⊺

A−1k ek,1 A−1

k

−

hk−1,1

4qk−1+(A−1k−1

)

11

(A−1k−1ek−1,1

)⊺p⊺

k,k−1

pk,k−1A−1k−1ek−1,1 pk,k−1A

−1k−1p

⊺

k,k−1

.

Observe that

hk,1qk

=h2k,1

ε=h2k−1,1

4ε=hk−1,1

4qk−1

and, hence, we obtain

Vk =

(A−1k

)

11−(A−1k−1

)

11

(A−1k ek,1

)⊺ −(A−1k−1ek−1,1

)⊺p⊺

k,k−1

A−1k ek,1 − pk,k−1A

−1k−1ek−1,1 A−1

k − pk,k−1A−1k−1p

⊺

k,k−1

.

Put Vk := A−1k − pk,k−1A

−1k−1p

⊺

k,k−1 and note that ek−1,1 = p⊺

k,k−1ek,1. Here, Vk is thematrix for the approximation property of the reduced Poisson problem, see (5.4). Thus,

Vk =

(

Vk

)

11

(

Vkek,1

)⊺

Vkek,1 Vk

.

We have to estimate ‖Vk‖0,1,k←0,1,k ≤ Cahk, i.e., the discrete Euclidean norm of

(

Vk

)

11qk

(

Vkek,1

)⊺

q−1k Vkek,1 Vk

has to been analysed. Since Vk is independent of qk we have to prove Vkek,1 = 0. Thenthe approximation property for problem (4.2) follows from the well-known approximationproperty for the reduced Poisson problem (5.4).

Lemma 5.7 Let ek,1 denote the first canonical unit vector in Rnk−1 and let Vk be defined

by Vk := A−1k − pk,k−1A

−1k−1p

⊺

k,k−1. Then

Vkek,1 = 0 .

5.3 Multigrid Convergence 47

Proof. Let us first compute the vector A−1k ek,1. Define the 2 × 2-matrix

m :=(

(ϕk,i, ϕk,j)L2(xk,1,xk,2)

)

1≤i,j≤2

and compute explicitly β := m−1(1, 0)⊺ = 2/hk,2(2,−1)⊺. Thus, the function f ∈ L2(ε, 1)

f(x) :=

β1ϕk,1 + β2ϕk,2 = 2(2xk,2 − 3x+ ε) in (xk,1, xk,2) ,0 otherwise

(5.16)

satisfies(∫ 1

ε fϕk,i dx)nk−1

i=1= ek,1. Define u as the solution of (5.4) with f as in (5.16)

u(x) :=

1∫

x

t∫

ε

f(s)dsdt=

(x−ε)2(x−xk,2−hk,2)

h2k,2

+ 1 − ε ε ≤ x ≤ xk,2 ,

1 − x xk,2 ≤ x ≤ 1 .(5.17)

It is well known that the finite element solution for the reduced one-dimensional Poissonproblem (5.4) is the nodal interpolant of the exact solution (see Lemma 1.10). Thus,

A−1k ek,1 = (u(xk,i))

nk−1i=1 = (1 − xk,i)

nk−1i=1 .

Next, we compute pk,k−1A−1k−1ek−1,1. From (5.17) we conclude

A−1k−1ek−1,1 = (1 − xk,i)

nk−1i=1

and by applying the definition of the grid points and the canonical finite element prolon-gation pk,k−1 we get

pk,k−1A−1k−1ek−1,1 = A−1

k ek,1 .

Thus, Vkek,1 = 0.Applying this result and the well-known approximation property for the reduced Poissonproblem (c.f. [Hac85, Proposition 6.3.14]) to Vk yields the approximation property forthe one-dimensional problem.

5.3 Multigrid Convergence

A combination of Lemma 5.3 and Lemma 5.4 yields the convergence of the twogrid methodwith respect to the ‖·‖0,1-norm uniform with respect to the overlap qk ∈ ]0, q] with q =O(1). From this convergence result, we easily derive the convergence of the W -cyclemultigrid method.

Theorem 5.8 There exist 0 < ω < ω < 1, 1 < ν < ν, and q > 0 independent of k suchthat

‖MGk‖0,1←0,1 ≤ 1/2 ∀ q ∈ ]0, q] , ν ∈ [ν, ν] , ω ∈ [ω, ω] ,

where MGk denotes the W-cycle multigrid iteration matrix with ν steps of the dampedJacobi method as the smoothing iteration.For q < q < 1, we have

‖MGk‖0←0 ≤ CMG (ν + 1)−1

for all ν ∈ N with CMG is a constant independent of qk, k, and the step size hk.


Proof. We adopt the theory and notations of [Hac85, Sec 7.1] and consider first the caseof a small overlap 0 < q ≤ q. We begin with the estimate of the ν-fold application of thesmoothing operator, i.e., ‖Kν

k‖0,1←0,1. Similarly as in the proof of the smoothing propertyin Lemma 5.3 we derive the splitting

Kνk = D

−1/2k N

−1/41,k (Ik − ωWk)

ν N1/41,kD

1/2k ,

with Wk := N1/41,kD

−1/2k LkD

−1/2k N

−1/41,k , which leads to

‖Kνk‖0,1←0,1 ≤

∥∥∥N

1/21,kD

−1/2k N

−1/41,k

∥∥∥

0←0‖(Ik − ωWk)

ν‖0←0

∥∥∥N

1/41,kD

1/2k N

−1/21,k

∥∥∥

0←0

≤ C ′‖Ik − ωWk‖ν0←0 .

The assumption 0 < ω ≤ ω ≤ 2/CW (cf. (5.14)) leads to

‖Kνk‖0,1←0,1 ≤ C = O(1)

and the constant CS in (3.16) satisfies CS = O(1). In order to estimate the constants Cpand Cp in (3.17) we have to investigate the eigenvalues of the symmetric matrix

N−1/21,k−1p

⊺

k,k−1N1,kpk,k−1N−1/21,k−1 =

14

14q 0 · · · 0

14q

14q

2 + 54

14

. . ....

0 14

32

. . . 0...

. . .. . .

. . . 14

0 · · · 0 14

32

.

By restricting to q ≤ 1/2, Gerschgorin’s theorem (c.f. Theorem A.1) implies that theeigenvalues of this matrix are bounded from below by 1/8 and from above by 2. Hence,Cp, Cp = O(1) and C⋆ := 1 + CpCp (1 + CS) = O(1) (cf. [Hac85, (7.1.6)]). Let MGk

denote the iteration matrix of the W -cycle multigrid method and Tk the iteration matrixof the twogrid method (cf. (5.1)). Then, according to [Hac85, (7.1.5c)] the recursion

‖MGk‖0,1←0,1 ≤ ‖Tk‖0,1←0,1 + C⋆‖MGk−1‖20,1←0,1

holds. Choose β in Lemma 5.3 as

β := (4C⋆CA)−1

and the bounds ν, ν, ω, ω, q = O(1) accordingly. Then, [Hac85, Lemma 7.1.6] implies that

‖MGk‖0,1←0,1 ≤ 1

2C⋆≤ 1/2

yielding the multigrid convergence with respect to the ‖·‖0,1-norm.Let us now consider the case q ∈ [q, 1[. Since q = O(1), the standard multigrid theory,e.g. [Hac85, Lemma 6.2.1, Prop. 6.2.14, Lemma 6.3.13], with respect to the ‖·‖0-normcan be applied yielding the asserted estimate. The constant in the estimate only dependson q = O(1).Summarising we have proved for a one-dimensional model problem that the multigridconvergence is robust with respect to a small overlap of elements with the domain by usinga weighted Euclidean norm. The analysis shows that this fact is related to the convergenceof unknowns corresponding to grid points lying (essentially) outside the domain. However,this is not considered as a drawback of the proposed method since these unknowns eitherare not interesting, since they are lying outside of the domain, or can be corrected easilyin a post-processing step via extrapolation from the interior.

6 Convergence Analysis for the

2D Model Problem

The convergence analysis in two dimensions is based on a partitioning of the domain. Thissplitting induces a partitioning of the system matrix Lk in order to separate the differentlyscaled entries in the system matrix.First, we analyse the convergence of a twogrid algorithm with respect to the levels k andk − 1. As the intergrid transfer operator we use the canonical finite element prolongation(see Definition 3.1) pk,k−1 : R

nk−1 → Rnk , and its transposed as the restriction rk−1,k.

The twogrid iteration matrix with ν (pre-)smoothing steps is given by

Tk(ν) :=(


−1k−1p

⊺

k−1,k

)

LkKνk .

The twogrid convergence is proved with respect to the Euclidean norm in the framework ofgeometric multigrid methods, i.e., by investigating the smoothing and the approximationproperty (c.f. Section 3.4.1).Recall that ε denotes the width of the strip Ω \ (0, 1) × (0, 1) (see Figure 4.5) and setqk := ε/hk. The matrix Dk is the diagonal part of Lk.

Theorem 6.1 For all 0 < β < 1 there exist 0 < ω < ω < 1, 1 < ν < ν and q > 0 suchthat for all qk ∈ ]0, q] , ν ∈ [ν, ν] , ω ∈ [ω, ω] :

‖D−1k LkK

νk‖0←0 ≤ β < 1 .

Theorem 6.1 is proved in an algebraic way using the partitioning of the system matrix.The proof is based on an expansion of each block of Lk in terms of qk (c.f. Section 6.2),in analogy to the proof of the smoothing property in one dimension.

Theorem 6.2 Choose q = O(1) as in Theorem 6.1. The approximation property holdsfor all qk ∈ ]0, q]

‖(L−1k − pk,k−1L

−1k−1p

⊺

k,k−1)Dk‖0←0 ≤ Ca

where Ca is independent of k and qk.

The proof of Theorem 6.2 is given in Section 6.3. The combination of these two resultsleads to the twogrid convergence: For all 0 < β < 1 we can fix bounds for the dampingfactor of the smoothing iteration ω and for the number of smoothing steps ν such that

‖Tk(ν)‖0←0 ≤ Caβ < 1 ,

where the constant Ca is independent of the scaling parameter qk and of the iteration levelk.

49


6.1 Splitting of the Domain

We split the domain Ω = (0, 1 + ε) × (0, 1) into two disjoint parts, the unit square ΩL :=(0, 1)2 and the small strip Ωε := (1, 1 + ε) × (0, 1) (cf. Figure 6.1).

r r r

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ΩL Ωε

ΓD ΓkΓN

τ1

τ2

τ3

τ4

τ5 τ9

(0, 0)

(1, 0)

(1, 0) (1 + ε, 0)

1 2 3

Figure 6.1: Initial grid on discretisation level k = 0. The domain Ω is split into twodisjoint parts ΩL and Ωε with Ω = ΩL ∪ Ωε.

This splitting leads to a partitioning of the system matrix Lk ∈ Rnk×nk :

Lk =

Ak B⊺

k

Bk Ck

, (6.1)

which we explain in detail below. Accordingly the domain Ωk which is covered by theoverlapping grid Gk is split for each discretisation level k. We define

GLk := τ ∈ Gk : τ ⊂ ΩL , GR

k := τ ∈ Gk : τ /∈ GLk , (6.2)

and

Ωk :=⋃

τ∈Gk

τ , ΩLk = ΩL , ΩR

k :=⋃

τ∈GRk

τ . (6.3)

Note that for all 0 ≤ k ≤ kmax it holds Ωε ⊂ ΩRk . Let

ΓN := 1 + ε × (0, 1) , ΓC := 1 × (0, 1) , Γk := 1 + hk × (0, 1) , (6.4)

where hk is defined by hk := 2−k−1. The sets of grid points for the two domains and theirinterface are given by

Θk :=

x = xk,ij = (ihk, jhk) : 1 ≤ i ≤ (2k+1 + 1), 1 ≤ j ≤ (2k+1 − 1)

,

ΘLk := Θk ∩ ΩL , ΘC

k := Θk ∩ ΓC , ΘRk := Θk \ (ΘL

k ∪ ΘCk ) .

(6.5)

Let nk = (2k+1 + 1)(2k+1 − 1) = ♯Θk, and we define ℓk := ♯ΘLk and rk := ♯ΘR

k = ♯ΘCk . We

use these notations to define the matrices in the representation of the partitioned system

6.1 Splitting of the Domain 51

matrix Lk in (6.1).

(Ak)ij :=

∫

ΩL

〈∇ϕk(xj),∇ϕk(xi)〉 , ∀xi, xj ∈ ΘLk ∪ ΘC

k , (6.6)

(Bk)ij :=

∫

Ωε

〈∇ϕk(xj),∇ϕk(xi)〉 , ∀xj ∈ ΘLk ∪ ΘC

k , xi ∈ ΘRk , (6.7)

(Ck)ij :=

∫

Ωε

〈∇ϕk(xj),∇ϕk(xi)〉 , ∀xi, xj ∈ ΘRk (6.8)

with Ak ∈ R(ℓk+rk)×(ℓk+rk), Bk ∈ R

rk×(ℓk+rk) and Ck ∈ Rrk×rk .

For the proofs of Theorem 6.1 and Theorem 6.2 we need an analogue partitioning of thediagonal part of the system matrix and an associated partitioning of the prolongationoperator. Define the diagonal matrices

DLk := diag Ak and DR

k := diag Ck . (6.9)

The canonical prolongation pk,k−1 : Rnk−1×nk−1 → R

nk×nk is partitioned into

pk,k−1 =:

[pLk,k−1 0

pCk,k−1 pR

k,k−1

]

(6.10)

with

pLk,k−1 ∈ R

(ℓk+rk)×(ℓk−1+rk−1) , pCk,k−1 ∈ R

rk×(ℓk−1+rk−1) , pRk,k−1 ∈ R

rk×rk−1 .

This way we have introduced local prolongation operators for each single block of Lk,acting between the discretisation levels k and k − 1.

6.1.1 Properties of the Single Block Matrices

We split the domain Ω in order to separate the differently scaled entries in the systemmatrix. All properties concerning the scaling factor qk can be deduced from the resultingsingle block matrices as will be shown below.

Definition 6.3 Define restriction operators RLk , RC

k , RRk and RLC

k for all vk ∈ Rnk and

wk ∈ Rℓk+rk by

RLk : R

nk → Rℓk+rk ; RL

kvk := ((vk)x)x∈ΘLk∪ΘC

k,

RCk : R

nk → Rrk ; RC

k vk := ((vk)x)x∈ΘCk,

RRk : R

nk → Rrk ; RR

k vk := ((vk)x)x∈ΘRk,

RLCk : R

ℓk+rk → Rrk ; RLC

k wk := ((wk)x)x∈ΘCk.

The extension operator ECLk : R

rk → Rℓk+rk is the transposed of the restriction operator

RLCk

(ECLk vk

)

x:=

(vk)x ∀x ∈ ΘC

k

0 else∀vk ∈ R

rk .


Remark 6.4 By using the definition of Bk in (6.7) we obtain, for each iteration level k,the matrix Bk has the explicit representation

Bk = −qkRLCk .

That means, all non-zero entries (Bk)ij are of order qk.

It is also possible to give an explicit representation of the inverse of matrix Ck.

Lemma 6.5 For qk ≤ 1/3, we have C−1k =: q−1

k Irk + QRRk with ‖QRR

k ‖0←0 ≤ 15.

Proof. Explicit calculations yield

Ck = qk

1 + qk − qk2 0 · · · 0

− qk2 1 + qk

. . .. . .

...

0. . .

. . . 0...

. . . 1 + qk − qk2

0 · · · 0 − qk2 1

= qk

(

Irk − qkQRRk

)

with

QRRk := −tridiag

[

−1

2, 1,−1

2

]

.

For qk ≤ 1/3 the Neumann series converges

C−1k = q−1

k

(∞∑

i=0

(

qkQRRk

)i)

= q−1k Irk + QRR

k

∞∑

i=0

(

qkQRRk

)i,

because ‖QRRk ‖0←0 ≤ 5/2. The norm of QRR

k := QRRk

∞∑

i=0

(

qkQRRk

)ican be estimated by

‖QRRk ‖0←0 ≤ ‖QRR

k ‖0←0

1 − qk‖QRRk ‖0←0

≤52

1 − 13 · 5

2

= 15 . (6.11)

Lemma 6.6 Let hk := 2−k−1. For 1 ≤ n ≤ 2k+1 − 1, the system

en := αn (sin (nπx2))(1,x2)∈ΘCk,

λn := − µn1 + qµn

with µn := 2 sin2 nπhk2

is an orthonormal eigensystem for the operator QRRk , where αn =

√2hk.

Proof. Explicit calculations yield

QRRk en = λn en .


The coefficients αn have to be chosen such that ‖en‖0 = 1:

‖en‖20 =

2k+1−1∑

j=1

(en)2j =

2k+1−1∑

j=1

α2n sin2(nπjhk) = α2

n

2k+1−1∑

j=1

sin2

(nπj

2k+1

)

.

The sum above can be simplified:

2k+1−1∑

j=1

sin2

(nπj

2k+1

)

=1

2

2k+1−1∑

j=1

(

1 − cos

(nπj

2k

))

=1

2hk− 1

2− 1

2

2k+1−1∑

j=1

cos

(nπj

2k

)

=1

2(h−1k − 1) − 1

2(−1)

=1

2h−1k ∀1 ≤ n ≤ 2k+1 − 1 .

Thus

‖en‖20 = α2

n

1

2h−1k .

We have for 1 ≤ n ≤ 2k+1 − 1 ‖en‖0 = 1: if αn =√

2hk.

Lemma 6.7 For qk ≤ 1/3, there holds C−1k Bk =: −RLC

k −qkQRCk where ‖QRC

k ‖0←0 ≤ 15.

Proof. This assertion is a direct conclusion of Remark 6.4 and Lemma 6.5, because

C−1k Bk = −IrkR

LCk − qkQ

RRk RLC

k = −RLCk − qkQ

RCk

with QRCk := QRR

k RLCk and, hence, ‖QRC

k ‖0←0 ≤ 15.Finally, we need estimates of the Euclidean norm of the diagonal part of the system matrix.

Remark 6.8 The explicit representation of the diagonal part of the partitioned systemmatrix is given by

DLk := diag Ak = diag [4, . . . , 4, 2 + 3qk − q2k, . . . , 2 + 3qk − q2k] ∈ R

(ℓk+rk)×(ℓk+rk) ,

DRk := diag Ck = diag [qk + q2k, . . . , qk + q2k] ∈ R

rk×rk .

Hence, the Euclidean norm of the diagonal matrices and their inverses is bounded by

‖DLk‖0←0 ≤ 4 , ‖(DL

k )−1‖0←0 ≤ 12+3qk−q

2k

,

‖DRk ‖0←0 ≤ qk + q2k , ‖(DR

k )−1‖0←0 ≤ 1qk+q2

k

.

6.1.2 Analysis of the Mass Matrix

The isomorphism Pk : Rnk → Sk(Ω) defined in (2.8) maps the coefficient vector vk to the

finite element function vk

vk = Pkvk =

nk∑

i=1

(v)iϕi|Ω .


The adjoint operator P ∗k : Sk(Ω) → Rnk is defined by

〈P ∗k u,v〉 = (u, Pkv)L2(Ω) ,

with the scalar product

〈u,v〉 =

nk∑

i=1

(u)i(v)i .

The product P ∗kPk defines the mass matrix Mk, i.e.

Mk := P ∗kPk , (Mk)ij =

∫

Ω

ϕk,j(x)ϕk,i(x) dx = (ϕk,j, ϕk,i)L2(Ω) . (6.12)

Lemma 6.9 There exists a diagonal matrix D such that

v⊺

kMkvk ≥ v⊺

kDkvk

for all vk ∈ Rnk .

Proof. An estimate of the mass matrix Mk can be derived from an estimate of the localmass matrix Mτ corresponding to each triangle τ ∈ Gk. A triangle τ ∈ Gk is characterisedby its edges xi, xj and xm, i.e. τ = τ(xi,xj ,xm).

v⊺

kMkvk = 〈Mkvk,vk〉 = (Pkvk, Pkvk)L2(Ω) = (vk, vk)L2(Ω) = ‖vk‖2L2(Ω)

=∑

τ∈Gk

‖vk‖2L2(τ) =

∑

τ(xi,xj ,xm)∈Gk

(vk,i, vk,j, vk,m)Mτ

vk,ivk,jvk,m

for any vk ∈ Rnk and v = (vk,i, vk,j, vk,m)⊺ ∈ R

3. For any triangle τ we determine afunction Cτ (qk, hk) such that v⊺Mτ v ≥ Cτ (qk, hk)v

⊺v. This estimate is equivalent to thenorm estimate ‖M−1

τ ‖0←0 ≤ C−1τ (qk, hk). There are three types of triangles, which appear

in the discretised model. The first one, called τ1, is completely covered by the domain Ω.The intersection between a triangle of second type τ2 and the domain has the measureε2/2. The overlap of Ω and the triangle τ3 is εhk − ε2/2. The three types of triangles aredepicted in Figure 6.2.The local mass matrices are defined by (Mτ )ij = (ϕk,j, ϕk,i)L2(τ). Explicit calculationslead to

Mτ1 =h2k

24

2 1 11 2 11 1 2

, Mτ2 = q2kh2k

12 − 2

3qk + 14q

2k

16qk − 1

8q2k

16qk − 1

8q2k

16qk − 1

8q2k

112q

2k

124q

2k

16qk − 1

8q2k

124q

2k

112q

2k

,

Mτ3 = qkh2k

13 − 1

2qk + 13q

2k − 1

12q3k

16 − 1

4qk + 16q

2k − 1

24q3k

14qk − 1

3q2k + 1

8q3k

16 − 1

4qk + 16q

2k − 1

24q3k

13 − 1

2qk + 13q

2k − 1

12q3k

14qk − 1

3q2k + 1

8q3k

14qk − 1

3q2k + 1

8q3k

14qk − 1

3q2k + 1

8q3k

13q

2k − 1

4q3k

.

Furthermore, the spectralnorm of the inverse of these local mass matrices satisfies

‖M−1τ1 ‖0←0 = 24h−2

k , ‖M−1τ2 ‖0←0 ≤ 24

q4kh2k

, ‖M−1τ3 ‖0←0 ≤ 24

q3kh2k


r r r

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

Ω

ΓD ΓN

τ1

τ3

τ2

(0, 0)

(0, 1)

(1, 0) (1 + ε, 0)

1 2 3

Figure 6.2: The three types of triangles depending on the overlap with the domain Ω.

and therefore,

Cτ1(qk, hk) =1

24h2k , Cτ2(qk, hk) =

q4kh2k

24, Cτ3(qk, hk) =

q3kh2k

24.

Now, going back from the local mass matrices to the global mass matrix the followingestimates are satisfied: For x ∈ ΘL

k it holds

∑

τ ∈ Gk

x is a vertex of τ

v⊺Mτ v ≥∑

τ ∈ Gk

x is a vertex of τ

Cτ1(qk, hk)v⊺v = 6 · 1

24h2kv

⊺v .

For x ∈ ΘCk it is satisfied

∑

τ ∈ Gk

x is a vertex of τ

v⊺Mτ v ≥∑

τ ∈ Gk

x is a vertex of τ

Cτ (qk, hk)v⊺v

= (3Cτ1(qk, hk) + 2Cτ3(qk, hk) + Cτ2(qk, hk)) v⊺v

=

(1

8h2k +

1

12q3kh

2k +

1

24q4kh

2k

)

v⊺v .

For x ∈ ΘRk we obtain

∑

τ ∈ Gk

x is a vertex of τ

v⊺Mτ v ≥∑

τ ∈ Gk

x is a vertex of τ

Cτ (qk, hk)v⊺v

= (2Cτ2(qk, hk) + Cτ3(qk, hk)) v⊺v

=

(1

24q3kh

2k +

1

12q4kh

2k

)

v⊺v .


Summarising, we get

v⊺

kMkvk =∑

τ∈Gk

v⊺Mτ v

≥ Cτ1(qk, hk)∑

τ1∈Gk

v⊺v +Cτ2(qk, hk)∑

τ2∈Gk

v⊺v + Cτ3(qk, hk)∑

τ3∈Gk

v⊺v

= Cτ1(qk, hk)

2k+1−1∑

i=1

3v2k,i + Cτ1(qk, hk)

2k+1(2k+1−1)∑

i=2k+1

(

3v2k,i + 3v2

k,i−2k+1+1

)

+ Cτ2(qk, hk)

nk∑

i=2k+1(2k+1−1)+1

(

2v2k,i + v2

k,i−2k+1+1

)

+ Cτ3(qk, hk)

nk∑

i=2k+1(2k+1−1)+1

(

v2k,i + 2v2

k,i−2k+1+1

)

=

ℓk∑

i=1

6 · Cτ1(qk, hk)v2k,i +

ℓk+rk∑

i=ℓk+1

(3Cτ1(qk, hk) + Cτ2(qk, hk) + 2Cτ3(qk, hk)) v2k,i

+

nk∑

i=ℓk+rk+1

(2Cτ2(qk, hk) + Cτ3(qk, hk)) v2k,i

=

nk∑

i=1

(Dk)iiv2k,i = v⊺

kDkvk

with

Dk :=1

4h2k diag

1, . . . , 1,︸︷︷︸

ℓk

1

2+

1

3q3k +

1

6q4k, . . . ,

1

2+

1

3q3k +

1

6q4k,

︸︷︷︸

rk

1

6q3k +

1

3q4k, . . . ,

1

6q3k +

1

3q4k

︸︷︷︸

rk

.

For the estimate of the inverse of the mass matrix we apply the Schur-complement repre-sentation.

Mk =:

[

AMk (BM

k )⊺

BMk CM

k

]

.

The single block matrices are given by

(AMk )ij :=

∫

ΩL

ϕk(xj)ϕk(xi) , ∀xi, xj ∈ ΘLk ∪ ΘC

k , (6.13)

(BMk )ij :=

∫

Ωε

ϕk(xj)ϕk(xi) , ∀xj ∈ ΘLk ∪ ΘC

k , xi ∈ ΘRk , (6.14)

(CMk )ij :=

∫

Ωε

ϕk(xj)ϕk(xi) , ∀xi, xj ∈ ΘRk (6.15)


Then, the Schur-complement representation is

M−1k =

(AMk )−1(Iℓk+rk + (BM

k )⊺(SMk )−1BM

k (AMk )−1) −(AM

k )−1(BMk )⊺(SM

k )−1

−(SMk )−1BM

k (AMk )−1 (SM

k )−1

,

with the Schur-complement SMk := CM

k − BMk (AM

k )−1(BMk )⊺.

Lemma 6.10 For the inverse of the mass matrix Mk defined by (6.12) it holds

‖(AMk )−1‖0←0 = O(h−2

k ) , ‖(BMk )‖0←0 = O(q2kh

2k) ,

‖(CMk )−1‖0←0 = O(q−3

k h−2k ) , and ‖(SM

k )−1‖0←0 = O(q−3k h−2

k )

Proof. The proof of Lemma 6.9 implies the following estimates

‖(AMk )−1‖0←0 = O(h−2

k ) , ‖(CMk )−1‖0←0 ≤ h−2

k

(q3k24

+q4k12

)−1

= O(q−3k h−2

k ) .

Furthermore, by using the block partitioning of the mass matrix and the definition of thediagonal matrix BM

k in (6.14) we obtain

‖BMk ‖0←0 = (Mτ3)32 + (Mτ2)21 =

1

4q2kh

2k −

1

6q3kh

2k = O(q2kh

2k) .

From this, it follows that ‖BMk (AM

k )−1(BMk )⊺(CM

k )−1‖0←0 = O(qk). By using this estimatewe obtain for the Euclidean norm of the inverse of the Schur-complement

‖(SMk )−1‖0←0 = ‖(CM

k )−1(Irk − BM

k (AMk )−1(BM

k )⊺(CMk )−1

)−1 ‖0←0

≤ ‖(CMk )−1‖0←0‖

(Irk − BM

k (AMk )−1(BM

k )⊺(CMk )−1

)−1 ‖0←0

≤ ‖(CMk )−1‖0←0 = O(q−3

k h−2k )

since qk < 1.

6.2 Proof of the Smoothing Property

This section is devoted to the proof of Theorem 6.1. There it is asserted that for 0 < β < 1there exist bounds for the damping factor of the smoothing iteration ω and for the numberof smoothing steps ν such that

‖D−1k L

kKνk‖0←0 ≤ β < 1 .

To simplify the notation we skip the index k in this section. Let W := D−1L. Then, weget

‖D−1LKν‖0←0 = ‖D−1L(I − ωD−1L

)ν ‖0←0

= ‖W (I − ωW)ν ‖0←0 .

We apply the partitioning (6.1) and the definitions (6.9) to obtain

W =

(DL)−1 0

0 (DR)−1

A B⊺

B C

=

(DL)−1A 0

(DR)−1B (DR)−1C

︸︷︷︸

=:WI

+

0 (DL)−1B⊺

0 0

︸︷︷︸

=:WII

. (6.16)


This leads to

W (I − ωW)ν = WI (I− ωWI − ωWII)ν + WII (I − ωW)ν . (6.17)

The first term in the right-hand side of equation (6.17) can be rewritten as a sum of termsof the form

WI

t∏

i=1

(I − ωWI)αi(−ωWII)

βi (6.18)

with αi, βi ∈ N0 and |α| + |β| = ν, |α| :=∑t

i=1 αi and |β| :=∑t

i=1 βi. There exists onlyone summand with |β| = 0, namely

WI(I − ωWI)ν .

In order to avoid the identity matrix as a factor in (6.18) we may assume, w.l.o.g.,

αi+1, βi > 0 ∀1 ≤ i ≤ t− 1 .

For the Euclidean norm of W(I − ωW)ν , we obtain the following estimate

‖W (I − ωW)ν ‖0←0 ≤ ‖WI (I− ωWI)ν‖0←0

+ν−1∑

j=0

(ν

j

)

‖WI‖0←0‖I − ωWI‖j0←0‖ωWII‖ν−j0←0

+

ν∑

j=0

(ν

j

)

‖WII‖0←0‖I − ωWI‖j0←0‖ωWII‖ν−j0←0

(6.19)

The estimation of the second and third term in (6.19) is reduced to the estimates of ‖WI‖,‖WII‖ and ‖I − ωWI‖:Lemma 6.11 There exist 0 < ω ≤ 1/2 and ν ≥ 1 such that for all 0 < q < 1, 0 < ω ≤ ωand 1 ≤ ν ≤ ν it holds

‖WI‖0←0 ≤ C1 ,

‖I − ωWI‖0←0 ≤ C2 ,

‖WII‖0←0 ≤ C3q .

The constants C1, C2 and C3 are independent of q and the iteration level k.

The proof of this lemma is given in Subsection 6.2.1. These estimates imply the followinglemma.

Lemma 6.12 For 0 < q < 1 and 0 < ω < 1 the following estimates for the sums in (6.19)are satisfied

ν−1∑

j=0

(ν

j

)

‖WI‖0←0‖I − ωWI‖j0←0‖ωWII‖ν−j0←0 ≤ν−1∑

j=0

(ν

j

)

Cj2(ωq)ν−j

= (C2 + ωq)ν − Cν2

andν∑

j=0

(ν

j

)

‖WII‖0←0‖I − ωWI‖j0←0‖ωWII‖ν−j0←0 ≤ν∑

j=0

(ν

j

)

qCj2(ωq)ν−j = q(C2 + ωq)ν .


Proof. With the assertions of Lemma 6.11 we conclude that

ν−1∑

j=0

(ν

j

)

‖WI‖0←0‖I − ωWI‖j0←0‖ωWII‖ν−j0←0 ≤ C1

ν−1∑

j=0

(ν

j

)

Cj2(C3ωq)ν−j

= C1 ((C2 + C3ωq)ν − Cν2 ) .

The first estimate in the right hand side of (6.19) ‖WI (I − ωWI)ν‖0←0 will be proved in

Subsection 6.2.2 and we state here only the result.

Lemma 6.13 For all 0 < ν ≤ ν ≤ ν and 1 < ω ≤ ω ≤ ω < 1 there exists a constant C5

such that

‖WI(I − ωWI)ν‖0←0 ≤ C5

ω(ν + 1),

where C5 is independent of q and h.

Combining this estimate with Lemma 6.12 results in

‖W(I − ωW)ν‖0←0 ≤ C5

ω(ν + 1)+ C1((C2 + C3ωq)

ν − Cν2 ) + q(C2 + ωq)ν . (6.20)

Next, we will determine the ranges of ω, ν and q such that every term in (6.20) is boundedby β/3 for any 0 < β < 1. Fix 0 < ω < ω ≤ 1/2 and 0 < β < 1. Put ν := 3C3ω

−1β−1 > 1and choose ν such that O(1) = ν > ν. Then,

‖W(I − ωW)ν‖0←0 ≤ β ∀q ∈]0, q], ν ∈ [ν, ν], ω ∈ [ω, ω]

with

q = min

C2

2 ω,

β

3 · 2ν ω C ν2

= O(1) .

For the calculation of q the following inequality has been used

(1 + a)n ≤ 1 + (2n − 1)a

for n ∈ N and 0 ≤ a ≤ 1. This yields the proof of the smoothing property with α = 0, seeTheorem 6.1 and Definition 3.7.

6.2.1 Estimates of ‖WI‖0←0, ‖WII‖0←0 and ‖I − ωWI‖0←0

The result of Remark 6.8 and (6.6) leads to the following norm estimate

‖(DL)−1A‖0←0 ≤ ‖(DL)−1‖0←0 ‖A‖∞ ≤ 8

2 + 3q − q2≤ C4 = O(1) . (6.21)

By using again Remark 6.8 and Remark 6.4 we obtain

‖(DR)−1B‖0←0 = supv∈Rℓ+r

‖(DR)−1Bv‖0

‖v‖0=

1

1 + q≤ 1 . (6.22)


In the next step we analyse the Euclidean norm of the matrix product (DR)−1C. Usingagain Remark 6.8 and the explicit representation of C given in the proof of Lemma 6.5results in

‖(DR)−1C‖0←0 =

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥

1 − −q2+2q 0 . . . 0

−q2+2q 1 −q

2+2q 0 . . . 0

0. . .

. . .. . .

. . ....

... −q2+2q 1 −q

2+2q 0

0 . . . 0 −q2+2q 1 −q

2+2q

0 . . . 0 −q2+2q 1

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥

0←0

≤ max

1 +q

2 + 2q, 1 +

q

1 + q

= 1 +q

1 + q≤ 3

2. (6.23)

For the last step we employed q < 1. Lemma A.3 and the definition of WI in (6.16) imply

‖WI‖20←0 ≤ 2

(‖(DL)−1A‖2

0←0 + ‖(DR)−1B‖20←0 + ‖(DR)−1C‖2

0←0

)

≤ 2

(

4 + 1 +9

4

)

< 15 .

Analogously we derive from Remark 6.8 and Remark 6.4

‖WII‖0←0 ≤√

2‖(DL)−1B⊺‖0←0 ≤ q

2. (6.24)

Now, ‖I − ωWI‖0←0 will be estimated by using the results (6.21) - (6.23), Lemma A.3and the Cauchy-Schwarz inequality

‖I − ωWI‖20←0 =

∥∥∥∥∥∥

I − ω(DL)−1A 0

−ω(DR)−1B I − ω(DR)−1C

∥∥∥∥∥∥

2

0←0

≤ 2(‖I − ω(DL)−1A‖2

0←0 + ω2‖(DR)−1B‖20←0 + ‖I − ω(DR)−1C‖2

0←0

)

≤ 2

(

2 + 16ω2 + ω2 +9

4ω2

)

≤ 4 +77

4ω2 .

Hence,

‖I − ωWI‖0←0 ≤ C2 ,

where the constant C2 is bounded by 4 for 0 < ω ≤ ω ≤ 1/2. These norm estimates areused in the proof of Lemma 6.12.

6.2.2 Estimate of ‖WI(I − ωWI)ν‖0←0

It remains to estimate ‖WI(I−ωWI)ν‖0←0. By using Lemma A.2 and explicit calculations

for the matrix (I − ωWI)ν we obtain

(I − ωWI)ν =

I − ω(DL)−1A 0

−ω(DR)−1B I − ω(DR)−1C

ν

=

(I − ω(DL)−1A)ν 0

T(ν) (I − ω(DR)−1C)ν


with T(ν) := −ω∑ν−1m=0

(I − ω(DR)−1C

)m(DR)−1B

(I − ω(DL)−1A

)ν−m−1. We define

A := (1 − ω)−1(I − ω(DL)−1A)

C := (1 − ω)−1(I − ω(DR)−1C)

T(ν) := (1 − ω)−1T(ν) .

(6.25)

Multiplication of (I − ωWI)ν with WI results in

WI(I − ωWI)ν = (1 − ω)ν

X(ν)I 0

X(ν)II X

(ν)III

,

where the block matrices are defined by

X(ν)I := (DL)−1AAν = (DL)−1A(1 − ω)−ν

(I − ω(DL)−1A

)ν,

X(ν)II := (DR)−1BAν + (DR)−1CT(ν) (6.26)

= (DR)−1B(1 − ω)−ν(I − ω(DL)−1A

)ν+ (DR)−1C(1 − ω)−1T(ν) and

X(ν)III := (DR)−1CCν = (DR)−1C(1 − ω)−ν

(I − ω(DR)−1C

)ν.

Applying again Lemma A.3 it follows

‖WI(I − ωWI)ν‖0←0 ≤

√2(1 − ω)ν

(

‖X(ν)I ‖2

0←0 + ‖X(ν)II ‖2

0←0 + ‖X(ν)III‖2

0←0

)1/2.

(6.27)

All summands will be estimated separately. We show, that for each block matrix X(ν)i ,

i ∈ I, II, III, there holds an estimate ‖X(ν)i ‖0←0 ≤ C(ω(ν + 1)(1 − ω)ν)−1. We start

with X(ν)I .

Lemma 6.14 Let DL be the diagonal part of A defined in (6.9). For 0 < ω ≤ ω ≤ 1/2,there holds

‖X(ν)I ‖0←0 = ‖(DL)−1AAν‖0←0 ≤

√2

ω(ν + 1)(1 − ω)ν.

Proof. For the proof of this lemma we apply the following splitting

‖(DL)−1A(I − ω(DL)−1A)ν‖0←0

= ‖(DL)−1A(DL)−1/2(I − ω(DL)−1/2A(DL)−1/2)ν(DL)1/2‖0←0

≤ ‖(DL)−1/2‖0←0‖(DL)1/2‖0←0‖(DL)−1/2A(DL)−1/2(I − ω(DL)−1/2A(DL)−1/2)ν‖0←0 .

A is symmetric (cf. (6.6)). Since therefore the matrix (DL)−1/2A(DL)−1/2 is symmetricits spectral radius equals to ‖(DL)−1/2A(DL)−1/2‖0←0. Thus, by using Remark 6.8 and(6.6) we have

‖(DL)−1/2A(DL)−1/2‖0←0 ≤ 8

2 + 3q − q2≤ C4 = O(1) .

The eigenvectors of I− ω(DL)−1/2A(DL)−1/2 coincide with those of (DL)−1/2A(DL)−1/2

and the eigenvalues of I − ω(DL)−1/2A(DL)−1/2 are of the form (1 − ωλ) with λ ∈


spec((DL)−1/2A(DL)−1/2

). Thus, the Euclidean norm of I − ω(DL)−1/2A(DL)−1/2 is

bounded by 1 for all 0 < ω ≤ ω ≤ 1/C4.By using Lemma 1.3.5 in [Hac85] we conclude that for any 0 < ω ≤ ω and ω ∈ [ω, ω],

‖(DL)−1/2A(DL)−1/2(I − ω(DL)−1/2A(DL)−1/2)ν‖0←0

≤ maxfν(λ) : λ ∈ spec((DL)−1/2A(DL)−1/2) .

with fν(λ) = λ(1 − ωλ)ν . The maximum above is bounded by

maxfν(λ) : λ ∈ spec((DL)−1/2A(DL)−1/2) ≤ fν

(1

ω(1 + ν)

)

,

i.e.,

‖(DL)−1/2A(DL)−1/2(I − ω(DL)−1/2A(DL)−1/2)ν‖0←0 ≤ νν

ω(ν + 1)ν+1

Therefore, by using Remark 6.8 we derive

‖X(ν)I ‖0←0 ≤

(λmax(D

L)

λmin(DL)

)1/2νν

ω(ν + 1)ν+1(1 − ω)ν≤

√2

ω(ν + 1)(1 − ω)ν.

In a similar way an estimate of the third summand in (6.27) is proved next.

Lemma 6.15 Let DR be the diagonal part of C defined in (6.9). For 0 < ω ≤ ω ≤ 1/2holds

‖X(ν)III‖0←0 = ‖(DR)−1CCν‖0←0 ≤ 1

ω(ν + 1)(1 − ω)ν.

Proof. For the proof of this lemma we apply the following splitting

‖(DR)−1C(I − ω(DR)−1C)ν‖0←0

= ‖(DR)−1C(DR)−1/2(I − ω(DR)−1/2C(DR)−1/2)ν(DR)1/2‖0←0

≤ ‖(DR)−1/2‖0←0‖(DR)1/2‖0←0‖(DR)−1/2C(DR)−1/2(I − ω(DR)−1/2C(DR)−1/2)ν‖0←0 .

C is symmetric (cf. (6.45)). Since therefore the matrix (DR)−1/2C(DR)−1/2 is symmetricits spectral radius equals ‖(DR)−1/2C(DR)−1/2‖0←0. Thus, by using Remark 6.8 and (6.8)we have

‖(DR)−1/2C(DR)−1/2‖0←0 ≤ q + 2q2

q + q2≤ 2 .

The eigenvectors of I−ω(DR)−1/2C(DR)−1/2 coincide with those of (DR)−1/2C(DR)−1/2

and the eigenvalues of I − ω(DR)−1/2C(DR)−1/2 are of the form (1 − ωλ) with λ ∈spec

((DR)−1/2C(DR)−1/2

). Thus, the Euclidean norm of I − ω(DR)−1/2C(DR)−1/2 is

bounded by 1 for all 0 < ω ≤ ω ≤ 1/2.From [Hac85, Lemma 1.3.5 ], we conclude that for any 0 < ω ≤ ω and ω ∈ [ω, ω],

‖(DR)−1/2C(DR)−1/2(I − ω(DR)−1/2C(DR)−1/2)ν‖≤ maxfν(λ) : λ ∈ spec((DR)−1/2C(DR)−1/2) .


with fν(λ) = λ(1 − ωλ)ν . The maximum above is bounded by

maxfν(λ) : λ ∈ spec((DR)−1/2C(DR)−1/2) ≤ fν

(1

ω(1 + ν)

)

,

i.e.,

‖(DR)−1/2C(DR)−1/2(I − ω(DR)−1/2C(DR)−1/2)ν‖0←0 ≤ νν

ω(ν + 1)ν+1

Finally, the Lemma is proved by employing again Remark 6.8:

‖X(ν)III‖0←0 ≤

(λmax(D

R)

λmin(DR)

)1/2νν

ω(ν + 1)ν+1(1 − ω)ν≤ 1

ω(ν + 1)(1 − ω)ν.

The choice ω ≤ 1/2 leads to the assertion.The combination of the result of Lemma 6.15 and an estimate of ‖A‖0←0 yields an estimate

of the Euclidean norm of the second summand of X(ν)II in (6.26).

Lemma 6.16 Let DR be the diagonal part of C and DL the diagonal part of A as in(6.9). For 0 < ω ≤ ω ≤ 1/2, there holds

‖X(ν)II ‖0←0 ≤ CνA

(

1 +2

ω3(ν + 1)(1 − ω)

)

.

Proof. The matrix X(ν)II has the explicit representation (cf. (6.26))

X(ν)II = (DR)−1BAν + (DR)−1CT(ν) =: X

(ν)II,1 + X

(ν)II,2 .

For the estimate of the Euclidean norm of the first summand X(ν)II,1 we use (6.21). Thus,

for ω ≤ ω < C−14 we have ‖A‖0←0 ≤ CA. Combining this result with (6.22) leads to

‖X(ν)II,1‖0←0 = ‖(DR)−1BAν‖0←0 ≤ CνA . (6.28)

Now, we give an explicit representation of the second summand X(ν)II,2

X(ν)II,2 = (DR)−1CT(ν) = (DR)−1C(−ω)(1 − ω)ν−2

ν−1∑

m=0

Cm(DR)−1BAν−m−1

= −ω(1 − ω)ν−2ν−1∑

m=0

X(m)IIIX

(ν−m−1)II,1

By using Lemma 6.15 and (6.28) we obtain

‖X(ν)II,2‖0←0 ≤ ω(1 − ω)ν−2

ν−1∑

m=0

‖X(m)III ‖0←0‖X(ν−m−1)

II,1 ‖0←0

≤ ω(1 − ω)ν−2ν−1∑

m=0

1

ω(m+ 1)(1 − ω)m‖A‖ν−m−1

0←0

≤ν−1∑

m=0

Cν−m−1A

(m+ 1)(1 − ω)ν−m−2 .


For all 0 < ω ≤ ω = O(1) there exists a maximal number of smoothing steps ν > 1 suchthat for all m ≤ ν it holds (1 + ω)m ≤ m+ 1. Further, let 1 < ν < ν be chosen such thatω(ν + 1) ≤ 1. Then it holds for all ν ≥ ν

ω(ν + 1) ≤ (1 + ω)ν .

By using these inequalities we obtain

‖X(ν)II,2‖0←0 ≤ 1

ω(ν + 1)(1 − ω)2

ν−1∑

m=0

ω(ν + 1)

m+ 1Cν−m−1A (1 − ω)ν−m

≤ 1

ω(ν + 1)(1 − ω)2

ν−1∑

m=0

Cν−m−1A

(1 + ω)ν

(1 + ω)m(1 − ω)ν−m

=1

ω(ν + 1)(1 − ω)2

ν−1∑

m=0

Cν−m−1A (1 − ω2)ν−m .

Assuming CA ≥ 1 we derive

‖X(ν)II,2‖0←0 ≤ Cν−1

A

ω(ν + 1)(1 − ω)2

ν−1∑

m=0

(1 − ω2)ν−m

=Cν−1A

ω(ν + 1)(1 − ω)2

ν∑

i=1

(1 − ω2)i

=Cν−1A

ω(ν + 1)(1 − ω)2

(ν∑

i=0

(1 − ω2)i − 1

)

=Cν−1A

ω(ν + 1)(1 − ω)2

(1 − (1 − ω2)ν+1

ω2− 1

)

=Cν−1A

(ν + 1)(1 − ω)21 − (1 − ω2)ν+1 − ω2

ω3

≤ Cν−1A (1 − ω2)

(ν + 1)(1 − ω)2ω3=

Cν−1A (1 + ω)

ω3(ν + 1)(1 − ω). (6.29)

Combining the results (6.28) and (6.29) yields the assertion.Summarising the results of Lemma 6.14, Lemma 6.15 and Lemma 6.16 lead to an estimateof the Euclidean norm of WI(I − ωWI)

ν (cf. (6.27):

‖WI(I − ωWI)ν‖0←0

≤√

2(1 − ω)ν

(

3

ω2(ν + 1)2(1 − ω)2ν+ C2ν

A

(

1 +2

ω3(ν + 1)(1 − ω)

)2)1/2

=

√2

ω(ν + 1)

√

3 +C2νA (1 − ω)2ν

(

ω(ν + 1) +2

ω2(1 − ω)

)2

.

Since we have chosen bounds 0 < ω < ω < 1 and 1 < ν < ν = O(1) we obtain theassertion of Lemma 6.13: For all ω ∈ [ω, ω] and ν ∈ [ν, ν] it holds

‖WI(I − ωWI)ν‖0←0 ≤ C5

ω(ν + 1).


6.3 Proof of the Approximation Property

Our aim (cf. (4.5)) is to prove

‖(L−1k − pk,k−1L

−1k−1p

⊺

k,k−1)Dk‖0←0 ≤ Ca (6.30)

with Ca independent of ε and of the dimension nk. To simplify the notation we skip theindex k if there is no ambiguity. In order to estimate the inverse of Lk we employ itsSchur-complement representation

L−1 =

A−1(I + B⊺S−1BA−1) −A−1B⊺S−1

−S−1BA−1 S−1

, (6.31)

with the Schur-complement S := C−BA−1B⊺. The proof of (6.30) and thus of Theorem6.2 is based on an expansion of L−1 in terms of q−1, which is derived for each blockseparately.We introduce the matrices V, VI , VII , VIII by

(


−1k−1p

⊺

k,k−1

)

Dk =: VDk =:

VIDLk V⊺

IIDRk

VIIDLk VIIID

Rk

. (6.32)

We will estimate the weighted norm of the coarse grid correction by means of Lemma A.3,i.e., by estimating the four blocks in (6.32) separately. The final results

1. ‖VIDLk‖0←0 ≤ 4C1 , 2. ‖VIID

Lk‖0←0 ≤ 4C2 ,

3. ‖V⊺

IIDRk ‖0←0 ≤ C2(qk + q2k) , 4. ‖VIIID

Rk ‖0←0 ≤ C3(1 + qk)

follow from

1. ‖VI‖0←0 ≤ C1 , 2. ‖VII‖0←0 ≤ C2 , 3. ‖VIII‖0←0 ≤ C3q−1k . (6.33)

by using ‖DLk‖0←0 = 4 and ‖DR

k ‖0←0 = qk + q2k (cf. Remark 6.8). These three estimatesare proved separately in the Subsections 6.3.1 - 6.3.4.We start with some definitions and notations. Let fk ∈ R

nk with (fk)x = 0 for all x ∈ΘLk ∪ ΘC

k . Define a finite element function fk ∈ Sk(Ω) by the following ansatz

fk =∑

x∈Θk

αk,xϕk,x|Ω , (6.34)

where the coefficients αk := (αk,x)x∈Θkare given as the solution of the linear equation

system Mkαk = fk, where Mk is the mass matrix as in (6.12). Note, that

fk = Pk(M−1k fk) ,

where Pk is defined by (2.8). Analogous, let gk ∈ Rnk with (gk)x = 0 for all x ∈ ΘR

k .Define a finite element function gk ∈ Sk(Ω) by the following ansatz

gk =∑

x∈Θk

βk,xϕk,x|Ω , (6.35)

where the coefficients βk := (βk,x)x∈Θkare given as the solution of the linear equation

system Mkβk = gk. Note, that

gk = Pk(M−1k gk) .


Remark 6.17 In general, the coefficients αk,x and βk,x are non-zero for all x ∈ Θk.

Let uk ∈ Sk(Ω) be the solution of the following problem: Seek a function uk ∈ Sk(Ω)which satisfies∫

Ω

〈∇uk,∇v〉 =

∫

Ω

fkv ∀v ∈ Sk(Ω) (6.36)

for given fk ∈ Sk(Ω). The associated coefficient vector is denoted uk ∈ Rnk .

Let uk−1 ∈ Sk−1(Ω) be the solution of the following problem: Seek a function uk−1 ∈Sk−1(Ω) which satisfies

∫

Ω

〈∇uk−1,∇v〉 =

∫

Ω

fkv ∀v ∈ Sk−1(Ω) (6.37)

for given fk ∈ Sk(Ω) and uk−1 is the associated coefficient vector.

Let uk ∈ Sk(Ω) be the solution of the following problem: Seek a function uk ∈ Sk(Ω)which satisfies∫

Ω

〈∇uk,∇v〉 =

∫

Ω

gkv ∀v ∈ Sk(Ω) (6.38)

for given gk ∈ Sk(Ω). The associated coefficient vector is denoted uk ∈ Rnk .

Let uk−1 ∈ Sk−1(Ω) be the solution of the following problem: Seek a function uk−1 ∈Sk−1(Ω) which satisfies

∫

Ω

〈∇uk−1,∇v〉 =

∫

Ω

gkv ∀v ∈ Sk−1(Ω) (6.39)

for given gk ∈ Sk(Ω) and uk−1 is the associated coefficient vector.

Lemma 6.18 Let fk ∈ Rnk with (fk)x = 0 for all x ∈ ΘL

k ∪ ΘCk and let gk ∈ R

nk with(gk)x = 0 for all x ∈ ΘR

k . Then the following two implications are satisfied

1. fk−1 := p⊺

k,k−1fk =

(∫

Ω

fkϕk−1,x

)

x∈Θk−1

=⇒ (fk−1)x = 0 ∀x ∈ ΘLk−1.

2. gk−1 := p⊺

k,k−1gk =

(∫

Ω

gkϕk−1,x

)

x∈Θk−1

=⇒ (gk−1)x = 0 ∀x ∈ ΘRk−1.

Proof. The definition of gk yields for all x ∈ Θk−1

(gk−1)x =∑

y∈Θk

(

p⊺

k,k−1

)

xy(gk)y =

∑

y∈Θk

(

p⊺

k,k−1

)

xy(gk, ϕk,y)L2(Ω)

=

gk,∑

y∈Θk

(

p⊺

k,k−1

)

xyϕk,y

L2(Ω)

=

gk,∑

y∈Θk

ϕk−1,x(y)ϕk,y

L2(Ω)

= (gk, ϕk−1,x)L2(Ω) .


Analogous calculations for the coefficients of vector fk−1 yield

(fk−1)x = (fk, ϕk−1,x)L2(Ω) .

By using the splitting of the prolongation matrix (6.10) we get

gk−1 := p⊺

k,k−1gk =

[

(pLk,k−1)

⊺ (pCk,k−1)

⊺

0 (pRk,k−1)

⊺

] [

RLkgk

0

]

=

[

(pLk,k−1)

⊺RLkgk

0

]

.

This implies the second assertion.For proving the first implication we proceed with

fk−1 := p⊺

k,k−1fk =

[

(pLk,k−1)

⊺ (pCk,k−1)

⊺

0 (pRk,k−1)

⊺

][

0

RRk fk

]

=

[

(pCk,k−1)

⊺RRk fk

(pRk,k−1)

⊺RRk fk

]

.

From the definition of(

pCk,k−1

)

ijit follows

(pCk,k−1

)

ij6= 0 ⇐⇒ ∃τ ∈ Gk, xi ∈ ΘR

k , xj ∈ ΘLk−1∪ΘC

k−1 such that xixj is an edge of τ .

Therefore, if 1 ≤ j ≤ ℓk then(

pCk,k−1

)

ij= 0 for all 1 ≤ i ≤ rk−1.

Hence, if 1 ≤ i ≤ ℓk−1 then((

pCk,k−1

)⊺)

ij= 0 for all 1 ≤ j ≤ rk.

This proves the first assertion.These preparatory considerations allow to transfer equations (6.36) - (6.39) into algebraicrelations

RLkL−1k gk = RL

k uk ,

RRk L−1

k gk = RRk uk ,

L−1k−1gk−1 = L−1

k−1p⊺

k,k−1gk = uk−1 ,

RRk L−1

k fk = RRk uk ,

L−1k−1fk−1 = L−1

k−1p⊺

k,k−1fk = uk−1 .

(6.40)

By using these relations the block matrices of the coarse grid correction V (c.f. (6.32))can be reformulated such that the explicit dependence of the block matrices on qk willfollow.

6.3.1 Estimate of ‖VI‖0←0

Let gk ∈ Rnk with (gk)x = 0 for all x ∈ ΘR

k . Using (6.40) we get

VIRLkgk = RL

k

(


−1k−1p

⊺

k,k−1

)

gk

= RLk (uk − pk,k−1uk−1)

= RLkP−1k (uk − uk−1) . (6.41)

Theorem 6.19 The Euclidean norm of the discrete operator VI is bounded from aboveby a constant, ‖VI‖0←0 ≤ CI .


Proof. We recall the definition of ‖ · ‖0 in (2.12)

‖u‖0 := ‖u‖0,Ω :=

∫

Ω

|u(x)|2 dx

1/2

∀u ∈ SCFEk (Ω) .

The assertion results from the proof of the following three estimates

1. ‖uk − uk−1‖0 ≤ C1h2k ‖gk‖0 ,

2. ‖RLkP−1k ‖0←0 = sup

vk∈Sk(Ω)\0

‖RLkP−1k vk‖0

‖vk‖0

≤ C2h−1k ,

3. supgk ∈ Rnk \ 0

suppgk ∈ ΘLk∪ΘC

k

‖gk‖0

‖RLkgk‖0

≤ C3h−1k .

These estimates are proved by the following three lemmata. Combining the first and thesecond estimate by using (6.41) we obtain

‖VIRLkgk‖0 ≤ ‖RL

kP−1k ‖0←0‖(uk − uk−1)‖0 ≤ C1C2hk‖gk‖0 .

By using the third estimate we derive the assertion.

Lemma 6.20 Let uk be the solution of problem (6.38) and uk−1 the solution of (6.39).It holds

‖uk − uk−1‖0 ≤ C1h2k ‖gk‖0 .

Proof. Let u be the exact solution of the problem: Seek a function u ∈ H1D(Ω) which

satisfies∫

Ω

〈∇u,∇v〉 =

∫

Ω

gkv ∀v ∈ H1D(Ω).

Applying the triangle inequality and the Aubin-Nitsche Lemma (c.f. A.8) result in

‖uk − uk−1‖0 ≤ ‖uk − u‖L2(Ω) + ‖u− uk−1‖L2(Ω)

≤ Ci‖uk − u‖H1(Ω) supψ∈L2(Ω)\0

1

‖ψ‖0

infv∈Sk(Ω)

‖uψ − v‖H1(Ω)

+ Cii‖u− uk−1‖H1(Ω) supψ∈L2(Ω)\0

1

‖ψ‖0

infv∈Sk−1(Ω)

‖uψ − v‖H1(Ω)

.

Furthermore the discrete solution u satisfies the quasi-optimality estimate

‖uk − u‖H1(Ω) ≤ Cqo infv∈Sk(Ω)

‖v − u‖H1(Ω) .

Therefore,

‖uk − uk−1‖L2(Ω)

≤ CiCqo infv∈Sk(Ω)

‖v − u‖H1(Ω) supψ∈L2(Ω)\0

1

‖ψ‖0

infv∈Sk(Ω)

‖uψ − v‖H1(Ω)

+ CiiCqo infv∈Sk−1(Ω)

‖v − uk−1‖H1(Ω) supψ∈L2(Ω)\0

1

‖ψ‖0

infv∈Sk−1(Ω)

‖uψ − v‖H1(Ω)

.


Standard interpolation error estimate (c.f. Theorem 1.16) leads to

infv∈Sk(Ωk)

‖v − u‖H1(Ω) ≤ Chk‖u‖H2(Ω) ,

where C is independent of u and hk. The associated operator of the bilinear a(·, ·) isH2-regular, for all gk ∈ Sk(Ω), and hence ‖u‖H2(Ω) ≤ Cr‖gk‖0 (c.f. inequality (1.20)).Summarising,

‖uk − uk−1‖0 ≤ Ciii hk‖gk‖0 supψ∈L2(Ω)\0

1

‖ψ‖0

hk‖ψ‖0

+ Civ hk−1‖gk−1‖0 supψ∈L2(Ω)\0

1

‖ψ‖0

hk−1‖ψ‖0

≤ C1h2k‖gk‖0 .

Lemma 6.21 Let Pk be the prolongation from Rnk onto Sk(Ω). Recall the restriction

operator RLk : R

nk → Rℓk+rk

RLkvk := ((vk)x)x∈ΘL

k∪ΘC

k

as in (6.3). Then,

‖RLkP−1k ‖0←0 ≤ C2h

−1k

Proof. We have Mk = (Pk)∗Pk. The assertion follows from estimates of the mass matrix

given in Subsection 6.1.2.

‖RLkP−1k ‖0←0 = sup

vk∈Sk(Ω)\0

‖RLkP−1k vk‖0

‖vk‖0

= supvk∈R

nk\0

‖RLkvk‖0

(Pkvk, Pkvk)1/2L2(Ω)

= supvk∈R

nk\0

‖RLkvk‖0

〈Mkvk,vk〉1/20

.

In Lemma 6.9 we have proved

v⊺

kMkvk ≥ v⊺

kDkvk

with Dk := 14h

2k diag

[1, . . . , 1, 1

2 + 13q

3k + 1

6q4k, . . . ,

12 + 1

3q3k + 1

6q4k,

16q

3k + 1

3q4k, . . . ,

16q

3k + 1

3q4k

].

By using this result we obtain

‖RLkP−1k ‖0←0 ≤ sup

vk∈Rnk\0

‖RLkvk‖0

〈Dkvk,vk〉1/20

= supvk∈Rnk\0

‖RLkvk‖0

〈D1/2k vk, D

1/2k vk〉1/20

= supvk∈Rnk\0

‖RLk D−1/2k vk‖0

‖vk‖0≤ 2

√2h−1

k .

Lemma 6.22 For all gk ∈ Rnk with (gk)x = 0 for all x ∈ ΘR

k there holds

‖gk‖0 ≤ C3h−1k ‖gk‖0 .


Proof. The definition of gk (cf. (6.35)) implies

supgk ∈ Rnk \ 0

suppgk ∈ ΘLk∪ΘC

k

‖gk‖0

‖RLkgk‖0

= supgk ∈ Rnk \ 0

supp gk ∈ ΘLk∪ΘC

k

(β⊺

kMkβk)1/2

‖RLkgk‖0

= supgk ∈ Rnk \ 0

supp gk ∈ ΘLk∪ΘC

k

〈M−1k gk,gk〉1/2‖RL

kgk‖0. (6.42)

We apply the Schur-complement representation for the inverse of the mass matrix

M =

[

AM (BM)⊺

BM CM

]

,

and, hence

M−1 =

(AM)−1(Iℓ+r + (BM)⊺(SM)−1BM(AM)−1) −(AM)−1(BM)⊺(SM)−1

−(SM)−1BM(AM)−1 (SM)−1

,

with the Schur-complement SM := CM−BM(AM)−1(BM)⊺. Because this matrix is appliedto a special right hand side vector g ∈ R

n with (g)x = 0 for all x ∈ ΘR for the scalarproduct 〈M−1g,g〉 it follows

〈M−1g,g〉 = g⊺M−1g

=(RLg

)⊺(AM)−1

(Iℓ+r + (BM)⊺(SM)−1BM(AM)−1

)RLg

≤ ‖(AM)−1‖0←0

(1 + ‖(BM)⊺(SM)−1BM‖0←0‖(AM)−1‖0←0

)‖RLg‖2

0 .

By using the results of Lemma 6.10

‖(AM)−1‖0←0 = O(h−2) , ‖(BM)‖0←0 = O(q2h2)

‖(CM)−1‖0←0 = O(q−3h−2) , and ‖(SM)−1‖0←0 = O(q−3h−2)

we obtain

〈M−1g,g〉 ≤ ‖(AM)−1‖0←0

(1 + ‖BM‖2

0←0‖(SM)−1‖0←0‖(AM)−1‖0←0

)‖RLg‖2

0

≤ Ch−2(1 + q)‖RLg‖20 ≤ C3h

−2‖RLg‖20 .

Inserting this result into (6.42) proves the assertion.

Combining the results of Lemma 6.20, Lemma 6.21 and Lemma 6.22 yields the mainestimate of this subsection ‖VI‖0←0 ≤ CI (cf. Theorem 6.19).

6.3.2 Partitioning of the system matrix

For the estimate of ‖VII‖0←0 and ‖VIII‖0←0 we need a refined partitioning of the systemmatrix Lk, which is based on the splitting of the domain Ω (c.f. Section 6.1).


We define

(ALk )ij :=

∫

ΩL

∇ϕk(xj)∇ϕk(xi) ∀xi, xj ∈ ΘLk and AL

k ∈ Rℓk×ℓk , (6.43)

(ACk )ij :=

∫

Ω

∇ϕk(xj)∇ϕk(xi) ∀xi, xj ∈ ΘCk and AC

k ∈ Rrk×rk , (6.44)

(CRk )ij :=

∫

Ωε

∇ϕk(xj)∇ϕk(xi) ∀xi, xj ∈ ΘRk and CR

k ∈ Rrk×rk , (6.45)

(AOk )ij :=

∫

ΩL

∇ϕk(xj)∇ϕk(xi) ∀xj ∈ ΘLk , xi ∈ ΘC

k and AOk ∈ R

rk×ℓk , (6.46)

(BOk )ij :=

∫

Ωε

∇ϕk(xj)∇ϕk(xi) ∀xj ∈ ΘCk , xi ∈ ΘR

k and BOk ∈ R

rk×rk , (6.47)

where ℓk is the dimension of ΘLk and rk = dim (ΘC

k ) = dim (ΘRk ). This leads to a parti-

tioning of the system matrix Lk

Lk =

ALk (AO

k )⊺ 0

AOk AC

k (BOk )⊺

0 BOk CR

k

. (6.48)

The associated block partitioning of the prolongation operator pk,k−1 ∈ Rnk×nk−1 is given

by

pk,k−1 =:

pLLk,k−1 pOO

k,k−1 0

0 pCCk,k−1 0

0 pORk,k−1 pRR

k,k−1

. (6.49)

6.3.3 Estimate of ‖VII‖0←0

This subsection is devoted to the proof of the following theorem:

Theorem 6.23 The Euclidean norm of the discrete operator VII is bounded from aboveby a constant, ‖VII‖0←0 ≤ CII .

Proof. Let gk ∈ Rnk with (gk)x = 0 for all x ∈ ΘR

k . To estimate its norm we firstreformulate VII as in the previous section using (6.40).

VIIRLkgk = RR

k

(


−1k−1p

⊺

k,k−1

)

gk

= RRk (uk − pk,k−1uk−1) . (6.50)

Applying the partitioning of the global system matrix Lk as in (6.48) yields

RRk uk =

(CRk

)−1 (RRk gk − BO

k RCk uk

),

RRk (pk,k−1uk−1) =

[pORk,k−1 pRR

k,k−1

]

RCk−1uk−1

(CRk−1

)−1(

RRk−1(p

⊺

k,k−1gk) − BOk−1R

Ck−1uk−1

)

.


Since RRk gk ≡ 0 one concludes RR

k−1(p⊺

k,k−1gk) ≡ 0 (c.f. Lemma 6.18) and the represen-tation above simplifies to

RRk uk = −

(CRk

)−1BOk RC

k uk ,

RRk (pk,k−1uk) =

(

[pORk,k−1 pRR

k,k−1

]

[Irk−1

−(CRk−1

)−1BOk−1

])

RCk−1uk−1 .

Hence, equation (6.50) becomes

VIIRLkgk = −

(CRk

)−1BOk RC

k uk −(

[pORk,k−1 pRR

k,k−1

]

[−Irk−1

(CRk−1

)−1BOk−1

])

RCk−1uk−1

.

This leads to the splitting

−VIIRLkgk =

(CRk

)−1BOk

︸︷︷︸

=:YI,k

(RCk uk − pCC

k,k−1RCk−1uk−1

)(6.51)

+

(CRk

)−1BOk pCC

k,k−1 −[pORk,k−1 pRR

k,k−1

]

[−Irk−1

(CRk−1

)−1BOk−1

]

︸︷︷︸

=:YII

RCk−1uk−1 .

The definitions of YI,k and YII yields the following representation of YII

YII = YI,kpCCk,k−1 −

[pORk,k−1 pRR

k,k−1

]

[

−Irk−1

YI,k−1

]

.

The assertion of this subsection ‖VII‖0←0 ≤ CII is proved by estimating the norm of bothsummands in (6.51) separately,

‖YI,k

(RCk uk − pCC


)‖0 ≤ C1‖gk‖0 and

‖YIIRCk−1uk−1‖0 ≤ C2qk‖gk‖0 .

The proofs of these two assertions are given by Lemma 6.24 and Lemma 6.27. Combiningthe two estimates and the triangle inequality for ‖ · ‖0 yield

‖VIIRLkgk‖0 ≤ ‖YI,k

(RCk uk − pCC


)‖0 + ‖YIIR

Ck−1uk−1‖0 ≤ C‖gk‖0

and thus the assertion.

Lemma 6.24 Let qk ≤ q < 1. It holds

‖YI,k

(RCk uk − pCC


)‖0 ≤ C‖gk‖0 .

Proof. By using the definitions (6.45) and (6.47) the following estimate is a simplecorollary of Lemma 6.7, which is proved in Subsection 6.1.1: Let qk ≤ 1/3. Then

‖YI,k‖0←0 = ‖(CRk

)−1BOk ‖0←0 ≤ C . (6.52)

Applying this result and the estimate

‖RCk uk − pCC

k,k−1RCk−1uk−1‖0 ≤ C‖gk‖0 ,

which is proved in Lemma 6.26, yields the assertion.


Lemma 6.25 1. For all uk ∈ Sk(Ω) it holds

‖uk‖0,ΓC ≤ C1h−1/2k ‖uk‖0,Ω .

2. For all gk ∈ Rnk with (gk)x = 0 for all x ∈ ΘR

k there holds

‖Pk(M−1k gk)‖0,Ω ≤ C2h

−1k ‖gk‖0 .

3. For all uk ∈ Sk(Ω), uk−1 ∈ Sk−1(Ω) and all gk ∈ Rnk with (gk)x = 0 for all x ∈ ΘR

k ,we have

‖uk − uk−1‖0,ΓC ≤ C3h1/2k ‖gk‖0 .

Proof.

ad 1. Applying standard scaling estimates for one-dimensional finite element spaces (c.f.for instance [Hac85, Subsection 6.3.1]) leads to the assertion

‖uk‖0,ΓC ≤ Ch1/2k ‖uk‖0 ≤ Ch

1/2k ‖ECL

k uk‖0 ≤ Ch−1/2k ‖uk‖0,Ω .

ad 2. The second assertion is a consequence of Lemma 6.22.

ad 3. Let u denote the exact solution of Problem 4.1. Then, standard finite element errorestimates (c.f. Theorem 1.19) yield

‖uk − uk−1‖0,Ω ≤ ‖uk − u‖0,Ω + ‖u− uk−1‖0,Ω ≤ Ch2k‖u‖H2(Ω) .

The bilinear form a(·, ·) is H2-regular and, therefore, we obtain

‖uk − uk−1‖0,Ω ≤ C · Crh2k‖gk‖0,Ω .

By using the second assertion, we get

‖uk − uk−1‖L2(Ω) ≤ Chk‖gk‖0 . (6.53)

The combination of Part 1. and (6.53) implies

‖uk − uk−1‖0,ΓC ≤ Ch−1/2k ‖uk − uk−1‖0,Ω ≤ Ch

1/2k ‖gk‖0 .

Lemma 6.26 Let uk be the associated coefficient vector of the solution of problem (6.38)and uk−1 the coefficient vector associated to the solution of problem (6.39). Then

‖RCk uk − pCC

k,k−1RCk−1uk−1‖0 ≤ C‖gk‖0 .

Proof. Let PCCk : R

rk → Sk(Ω)|ΓC be the one-dimensional canonical prolongation alongthe interface ΓC:

PCCk v =

∑

x∈ΘCk

(v)x (ϕx,k|ΓC) ∀v ∈ Rrk .


Since PCCk is a one-dimensional prolongation operator along the interface ΓC, we have

∥∥PCC

k

∥∥

0,ΓC←0≥ Ch

1/2k .

Hence, with Lemma 6.25 it follows

‖RCk uk − pCC

k,k−1RCk−1uk−1‖0 ≤ Ch

−1/2k ‖uk − uk−1‖0,ΓC ≤ C‖gk‖0 .

Lemma 6.27 The norm of YIIRCk−1uk−1 is bounded from above by the norm of the

right-hand side vector gk times qk:

‖YIIRCk−1uk−1‖0 ≤ C2qk‖gk‖0 .

Proof. Recall the definition of YII

YII :=(CRk

)−1BOk pCC


k,k−1

]

[−Irk−1

(CRk−1

)−1BOk−1

]

.

By using Lemma 6.5 and the definitions of BOk and CR

k in (6.47) and (6.45) we obtainBOk = −qkIrk and CR

k = Ck, and thus, (CRk )−1BO

k = −Irk − qkQRRk with ‖QRR

k ‖0←0 ≤ 15.This leads to the representation

YIIRCk−1uk−1 =

(−Irkp

CCk,k−1 + pOR

k,k−1Irk−1+ pRR

k,k−1Irk−1

−qkQRRk pCC

k,k−1 + qk−1pRRk,k−1Q

RRk−1

)RCk−1uk−1 .

Applying the definition of the partitioned prolongation matrix in (6.49) explicit calculationleads to

−IrkpCCk,k−1 + pOR

k,k−1Irk−1+ pRR

k,k−1Irk−1= 0

and hence,

YIIRCk−1uk−1 = −

(qkQ

RRk pCC

k,k−1 − qk−1pRRk,k−1Q

RRk−1

)RCk−1uk−1 .

Now, RCk−1uk−1 = RC

k−1L−1k−1gk−1 (see (6.39), where the restriction operator RC

k is as inby Definition 6.3. The right hand side vector gk−1 = p⊺

k,k−1gk with (gk−1)x = 0 for all

x ∈ ΘRk−1 (c.f. Lemma 6.18). The estimate of YIIR

Ck−1L

−1k−1gk−1 is split in the following

two estimates which are considered below∥∥∥qkQ

RRk pCC

k,k−1RCk−1L

−1k−1p

⊺

k,k−1gk

∥∥∥

0≤ Cqk‖gk‖0 , (6.54)

∥∥∥qk−1p

RRk,k−1Q

RRk−1R

Ck−1L

−1k−1p

⊺

k,k−1gk

∥∥∥

0≤ Cqk‖gk‖0 . (6.55)

Next we will show

∥∥QRR

k RCk L−1

k

∥∥

0←0≤ C . (6.56)

Then, the second estimate (6.55) follows from (6.56) by replacing k by k− 1 and by usingthe estimates ‖gk−1‖0 ≤ C‖gk‖0 and ‖pRR

k,k−1‖0←0 ≤ 2 (see Remark 3.4).


The second one (6.54) follows from (6.56), the estimate ‖QRRk ‖0←0 ≤ 15, which is proved

in Lemma 6.5, and Lemma 6.26 by using

∥∥∥

(

RCk L−1

k − pCCk,k−1R

Ck−1L

−1k−1p

⊺

k,k−1

)

gk

∥∥∥

0≤ C‖gk‖0

via a triangle inequality

∥∥∥QRR

k pCCk,k−1R

Ck−1L

−1k−1p

⊺

k,k−1gk

∥∥∥

0

≤∥∥QRR

k RCk L−1

k

∥∥

0←0‖gk‖0

+∥∥QRR

k

∥∥

0←0

∥∥∥

(

RCk L−1

k − pCCk,k−1R

Ck−1L

−1k−1p

⊺

k,k−1

)

gk

∥∥∥

0.

It remains to prove (6.56). Numerical tests in Appendix B show that the estimate‖QRR

k RCk L−1

k ‖0←0 ≤ ‖QRRk ‖0←0‖RC

k L−1k ‖0←0 is not sharp enough. In order to estimate

the norm of the product QRRk RC

k L−1k we first analyse the structure of the eigenvalues of

Lk.

Lemma 6.28 Let hk := 2−k−1. For 1 ≤ n ≤ 2k+1−1 and 1 ≤ m ≤ 2k+1, the eigensystemof L is given by

em,n(x) = αm,n

sin (ξmx1) sin (nπx2) x = (x1, x2) ∈ Θk \ ΘR

k ,γm,n sin (nπx2) x = (x1, x2) ∈ ΘR

k ,

λm,n := 4

(

sin2 ξmhk2

+ sin2 nπhk2

)

,

where ξm is the solution of

q (γm,n cos(nπh) − sin(ξm)) = 4

(

sin2

(ξmh

2

)

+ sin2

(nπh

2

))

γm,n (6.57)

with

γm,n =q − 1

q

(

1 − 2(q − 1) sin2

(nπh

2

))

sin(ξm) +sin(ξm(1 + h))

q.

Since L is symmetric and positive definite, the eigenvectors generate an orthogonal system.The scaling factor αm,n is chosen such that ‖em,n‖0 = 1.

Proof. For x ∈ ΘL the equation (Lem,n)x = λm,n(em,n)x has the representation

(Lem,n)x = 4 sin(ξmx1) sin(nπx2)

−∑

σ∈+,−

sin (ξm(x1 + σh)) · sin(nπx2) + sin(ξmx1) · sin (nπ(x2 + σh))

= 4

(

sin2

(ξmh

2

)

+ sin2

(nπh

2

))

sin(ξmx1) · sin(nπx2) .

That means, the eigenvalue associated to this eigenvector has the representation

λm,n = 4

(

sin2

(ξmh

2

)

+ sin2

(nπh

2

))

.


Next, we consider (Lem,n)x = λm,n(em,n)x for x = (x1, x2) = (1, x2) ∈ ΘC:

(Lem,n)x = (2 + 3q − q2) sin(ξmx1) · sin(nπx2) − sin (ξm(x1 − h)) · sin(nπx2)

−∑

σ∈+,−

(1

2+ q − q2

2

)

sin(ξmx1) · sin (nπ(x2 + σh)) − qγm,n sin(nπx2)

!= λm,n sin(ξmx1) · sin(nπx2) .

Hence, we get after multiple use of trigonometric identities

γm,n =q − 1

q

(

1 − 2(q − 1) sin2

(nπh

2

))


q.

Finally, we have to determine ξm. Therefor we apply the equations (Lem,n)x = λm,n(em,n)xfor x = (x1, x2) = (1 + h, x2) ∈ ΘR:

(Lem,n)x

= q(q + 1) γm,n sin(nπx2) −q2

2γm,n

∑

σ∈+,−

sin (nπ(x2 + σh)) − q sin(ξm) sin(nπx2)

= q (γm,n cos(nπh) − sin(ξm)) sin(nπx2)

!= 4

(

sin2 ξmhk2

+ sin2 nπhk2

)

γm,n sin(nπx2) .

Hence, the conditional equation for ξm is

q (γm,n cos(nπh) − sin(ξm)) = 4

(

sin2

(ξmh

2

)

+ sin2

(nπh

2

))

γm,n

with

γm,n =q − 1

q

(

1 − 2(q − 1) sin2

(nπh

2

))


q.

Let RCk be the restriction of a vector v ∈ R

nk on the grid points , which lie on ΓC (seeDefinition 6.3). Then, it holds

QRRk RC

kL−1k v = QRR

k

2k+1−1∑

n=1

2k+1∑

m=1

RCk L−1

k vm,nem,n = QRRk

2k+1−1∑

n=1

2k+1∑

m=1

vm,nλm,n

RCk em,n

= QRRk

2k+1−1∑

n=1

2k+1∑

m=1

vm,nαm,nλm,nαn

en

with vm,n := 〈v, em,n〉 and en = αn (sin(nπx2))(1,x2)∈ΘCk. By using Lemma 6.6 we obtain

QRRk en = λnen

with

λn = − µn1 + µn

and µn = 2 sin2

(nπhk

2

)


and the coefficients αn satisfy α2n = 2hk. Therefore, we get

|λn|αn

≤ Ch−1/2k .

From this it follows that

‖QRRk RC

k L−1k v‖2

0 =

∥∥∥∥∥∥

2k+1−1∑

n=1

2k+1∑

m=1

vm,nαm,nλm,nαn

λnen

∥∥∥∥∥∥

2

0

=

2k+1−1∑

n=1

λ2n

α2n

2k+1∑

m=1

vm,nαm,nλm,n

2

.

(6.58)

By using the Cauchy-Schwarz inequality we get

‖QRRk RC

k L−1k v‖2

0 ≤2k+1−1∑

n=1

λ2n

α2n

2k+1∑

m=1

α2m,n

λ2m,n

2k+1∑

m=1

v2m,n

≤ max1≤n≤2k+1−1

λ2n

α2n

2k+1∑

m=1

α2m,n

λ2m,n

‖v‖2

0

and finally

‖QRRk RC

k L−1k ‖0←0 ≤ max

1≤n≤2k+1−1

|λn|αn

√√√√

2k+1∑

m=1

α2m,n

λ2m,n

. (6.59)

We omit the very technical analysis of the transcendental equation (6.57), i.e., the depen-dence of ξm on qk and hk to estimate the right hand side in (6.59) explicitely, but replacethis step by numerical experiments, which are reported in Table 6.1 (see also AppendixB). These results strongly support the following heuristics:

‖QRRk RC

k L−1k ‖0←0 ≤ C , (6.60)

where C is independent of qk, hk and k.

k‖QRRk RC

k L−1k ‖0←0 1 2 3 4 5

2−2 0.472 0.472 0.473 0.474 0.4742−4 0.770 0.778 0.779 0.779 0.7792−6 0.912 0.935 0.940 0.941 0.9422−8 0.954 0.982 0.988 0.990 0.990

qk 2−10 0.965 0.994 1.001 1.003 1.0062−12 0.968 0.997 1.004 1.006 1.0062−14 0.969 0.998 1.005 1.007 1.0072−16 0.969 0.998 1.005 1.007 1.007

Table 6.1: ‖QRRk RC

kL−1k ‖0←0 is bounded by a constant C.


6.3.4 Estimate of ‖VIII‖0←0

The main assertion of this subsection is given in the following theorem.

Theorem 6.29 The Euclidean norm of the discrete operator VIII as in (6.32) is boundedfrom above by a constant times q−1

k , i.e., ‖VIII‖0←0 ≤ CIIIq−1k .

Proof. Let fk ∈ Rnk be given such that (fk)x = 0 for all x ∈ ΘL

k ∪ ΘCk . Again we apply

(6.40) to reformulate VIII .

VIIIRRk fk = RR

k

(


−1k−1p

⊺

k,k−1

)

fk

= RRk (uk − pk,k−1uk−1) . (6.61)

Applying the partitioning of the global system matrix Lk given in (6.48) and the parti-tioned prolongation operator (6.49) yields

RRk uk =

(CRk

)−1 (RRk fk − BO

k RCk uk

)

and

RRk (pk,k−1uk−1) = pOR

k,k−1RCk−1uk−1 + pRR

k,k−1RRk−1uk−1

= pRRk,k−1

(CRk−1

)−1 (RRk−1fk−1 − BO

k−1RCk−1uk−1

)+ pOR


= pRRk,k−1

(CRk−1

)−1RRk−1fk−1 +

(

[pORk,k−1 pRR

k,k−1

]

[Irk−1

−(CRk−1

)−1BOk−1

])

RCk−1uk−1 .

We define the matrix T by

T :=(CRk

)−1 − pRRk,k−1

(CRk−1

)−1 (pRRk,k−1

)⊺(6.62)

Then equation (6.61) becomes (with YI,k as defined in (6.51))

VIIIRRk fk = TRR

k fk − YI,k

(RCk uk − pCC


)

−

YI,kpCCk,k−1 −

[pORk,k−1 pRR

k,k−1

]

[

−Irk−1

YI,k−1

]

︸︷︷︸

=:YII

RCk−1uk−1 . (6.63)

The assertion of this subsection ‖VIII‖0←0 ≤ CIIIq−1k follows from estimates of the norm

of the three summands in (6.63) separately (see Lemma 6.30 - Lemma 6.32). Combiningthe results of these lemmata

‖TRRk fk‖0 ≤ C1q

−1k ‖fk‖0 ,

‖YI,k

(RCk uk − pCC


)‖0 ≤ C2‖f‖0 ,

‖YIIRCk−1uk−1‖0 ≤ C3‖f‖0

yields the assertion via a triangle inequality:

‖VIIIRRk fk‖0 ≤ (C1 + C2 + C3qk)q

−1k ‖fk‖0 .


Lemma 6.30 Let T be as in (6.62). Then

‖TRRk fk‖0 ≤ C1q

−1k ‖fk‖0 .

Proof. Applying the result of Lemma 6.5 and qk = 2qk−1 it follows

T = q−1k

(

Irk − 2pRRk,k−1Irk−1

(pRRk,k−1

)⊺)

+(

QRRk − pRR

k,k−1QRRk−1

(pRRk,k−1

)⊺)

.

By using the norm estimate∥∥QRR

k

∥∥

0←0≤ 15 (cf. (6.11)) we obtain via a triangle inequality

‖T‖0←0 ≤ q−1k

∥∥∥Irk − 2pRR

k,k−1Irk−1

(pRRk,k−1

)⊺∥∥∥

0←0+∥∥∥QRR

k − pRRk,k−1Q

RRk−1

(pRRk,k−1

)⊺∥∥∥

0←0

≤ q−1k + 2q−1

k ‖pRRk,k−1‖0←0‖(pRR

k,k−1)⊺‖0←0 + 15

(1 + ‖pRR

k,k−1‖0←0‖(pRRk,k−1)

⊺‖0←0

)

= q−1k + 15 + ‖pRR

k,k−1‖0←0‖(pRRk,k−1)

⊺‖0←0(2q−1k + 15) .

Remark 3.4 implies ‖pRRk,k−1‖0←0 ≤ 2 and ‖(pRR

k,k−1)⊺‖0←0 ≤ 2. We get the assertion via

the estimate

‖TRRk fk‖0 ≤ ‖T‖0←0‖RR

k fk‖0 = ‖T‖0←0‖fk‖0 ≤(15 + q−1

k + 4(15 + 2q−1k ))‖fk‖0 .

We turn now to the second summand in (6.63). Since the eigenfunctions of Lk are known(see Lemma 6.28) a spectral analysis of the operator RC

kL−1k − pCC

k,k−1RCk−1L

−1k−1p

⊺

k,k−1

can be performed as in (6.58). Again the problem boils down to the analysis of thetranscendental equation (6.57). As in (6.60) we replace this step by numerical experiments,which are reported in Table 6.2. These results strongly support the following hypothesis∥∥∥RC

kL−1k − pCC

k,k−1RCk−1L

−1k−1p

⊺

k,k−1

∥∥∥

0←0≤ C . (6.64)

k∥∥∥RC

kL−1k − pCC

k,k−1RCk−1L

−1k−1p

⊺

k,k−1

∥∥∥

0←0 1 2 3 4 5

2−2 0.593 0.593 0.593 0.593 0.5932−4 0.757 0.757 0.757 0.757 0.8352−6 0.814 0.814 0.814 0.814 0.8352−8 0.829 0.830 0.830 0.830 0.835

qk 2−10 0.833 0.834 0.834 0.835 0.8352−12 0.834 0.835 0.835 0.835 0.8352−14 0.834 0.835 0.835 0.835 0.8352−16 0.834 0.835 0.835 0.835 0.835

Table 6.2:∥∥∥RC

k L−1k − pCC

k,k−1RCk−1L

−1k−1p

⊺

k,k−1

∥∥∥

0←0is bounded by a constant C.

Lemma 6.31 Assume (6.64) holds. Let qk ≤ q < 1. It holds

‖YI,k

(RCk uk − pCC


)‖0 ≤ C2‖fk‖0 .


Proof. Let uk be the associated coefficient vector of the solution of problem (6.36) andlet uk−1 denote the coefficient vector associated to the solution of problem (6.37). Then,RCk−1uk−1 = RC

k−1L−1k−1fk−1, where the restriction operator RC

k−1 is defined in (3.3) and

fk−1 = p⊺

k,k−1fk and with (fk−1)x = 0 for all x ∈ ΘLk−1 (c.f. Lemma 6.18). Then,

RCk uk − pCC

k,k−1RCk−1uk−1 = RC

kL−1k fk − pCC

k,k−1RCk−1L

−1k−1fk−1

=(

RCk L−1

k − pCCk,k−1R

Ck−1L

−1k−1p

⊺

k,k−1

)

fk .

The assertion follows by combining (6.64) and (6.52),

‖YI,k

(RCk uk − pCC


)‖0

≤ ‖YI,k‖0←0

∥∥∥RC

k L−1k − pCC

k,k−1RCk−1L

−1k−1p

⊺

k,k−1

∥∥∥

0←0‖fk‖0 .

Lemma 6.32 Assume (6.60) and (6.64) hold. The norm of YIIRCk−1uk−1 is bounded

from above by the norm of the right-hand side vector fk:

‖YIIRCk−1uk−1‖0 ≤ C3‖fk‖0 .

Proof. Recall the definition of YII

YII :=(CRk

)−1BOk pCC


k,k−1

]

[−Irk−1

(CRk−1

)−1BOk−1

]

.

By using Lemma 6.5 and the Definitions of BOk and CR

k in (6.47) and (6.45) we obtainBOk = −qkIrk and CR

k = Ck, and thus, (CRk )−1BO

k = −Irk − qkQRRk with ‖QRR

k ‖0←0 ≤ 15.This leads to the representation

YIIRCk−1uk−1 = −

(qkQ

RRk pCC

k,k−1 − qk−1pRRk,k−1Q

RRk−1

)RCk−1L

−1k−1fk−1 .

The estimate of the spectral norm of YIIRCk−1L

−1k−1fk−1 is split into the following two

estimates which are proved in the sequel∥∥∥qkQ

RRk pCC

k,k−1RCk−1L

−1k−1p

⊺

k,k−1fk

∥∥∥

0≤ C‖fk‖0 , (6.65)

∥∥∥qk−1p

RRk,k−1Q

RRk−1R

Ck−1L

−1k−1p

⊺

k,k−1fk

∥∥∥

0≤ Cqk‖fk‖0 . (6.66)

Note that the estimates differ from the proof of Lemma 6.27 because of the differentright-hand sides fk and gk. The second estimate (6.66) follows from (6.60)

∥∥QRR

k RCk L−1

k

∥∥

0←0≤ C

by replacing k by k − 1 and the estimates ‖fk−1‖0 ≤ C‖fk‖0 and ‖pRRk,k−1‖0←0 ≤ 2 (see

Remark 3.4).We employ (6.60) once more: ‖QRR

k RCk L−1

k ‖0←0 ≤ C. Then, the first estimate (6.65)follows from the estimate ‖QRR

k ‖0←0 ≤ 15, which is proved in Lemma 6.7, and (6.64)

∥∥∥

(

RLCk L−1

k − pCCk,k−1R

Ck−1L

−1k−1p

⊺

k,k−1

)

fk

∥∥∥

0≤ Cq−1

k ‖fk‖0


via a triangle inequality

∥∥∥QRR

k pCCk,k−1R

Ck−1L

−1k−1p

⊺

k,k−1fk

∥∥∥

0

≤∥∥QRR

k RCk L−1

k

∥∥

0←0‖fk‖0

+∥∥QRR

k

∥∥

0←0

∥∥∥

(

RCk L−1

k − pCCk,k−1R

Ck−1L

−1k−1p

⊺

k,k−1

)

fk

∥∥∥

0.

6.4 Multigrid Convergence

A combination of Theorem 6.1 and Theorem 6.2 yields the convergence of the twogridmethod with respect to the Euclidean norm uniform with respect to the overlap q ∈ ]0, q]with q = O(1). In the case of q ∈ [q, 1[ the standard convergence theory, e.g. [Hac85,Lemma 6.2.1, Prop. 6.2.14, Theorem 6.3.15], can be applied.

Theorem 6.33 1. For all 0 < β < 1 there exist 0 < ω < ω < 1, 1 < ν < ν, and q > 0independent of k such that

‖Tk(ν)‖0←0 ≤ Caβ < 1 ,

where the constant Ca is independent of the scaling parameter qk and of the iterationlevel k.

2. For qk ∈ [q, 1[ holds

‖Tk(ν)‖0←0 ≤ Ca(ν + 1)−1 .

Proof.

ad 1. By using Theorem 6.1 and Theorem 6.2 we obtain the assertion.

ad 2. Since q = O(1) the standard twogrid convergence theory, e.g. [Hac85, Chapter 6],with respect to the ‖ · ‖0-norm can be applied yielding the asserted estimate: Acombination of Lemma 6.2.1 and Proposition 6.2.14 yields the smoothing property

‖LkKνk‖0←0 ≤ 1

ν + 1.

The approximation property arises from Theorem 6.3.15 by using the following es-timate

Cp−1‖uk‖0 ≤ ‖Pkuk‖0 ≤ Cp‖uk‖0 ∀uk ∈ Sk(Ω) .

This estimate is satisfied with Cp = 1 and Cp = 4 because of Lemma 6.9, since

‖Pkuk‖0 = (Pk,uk, Pkuk)1/2L2(Ω)

= 〈Mkuk,uk〉1/20 .

From this convergence result, we derive the convergence of the W -cycle multigrid method.


Theorem 6.34 1. There exist 0 < ω < ω < 1, 1 < ν < ν, and q > 0 independent of ksuch that

‖MGk‖0←0 ≤ 1/2 ∀ q ∈ ]0, q] , ν ∈ [ν, ν] , ω ∈ [ω, ω] ,

where MGk denotes the W-cycle multigrid iteration matrix with ν steps of thedamped Jacobi method as the smoothing iteration.

2. For q < q < 1, we have

‖MGk‖0←0 ≤ CMG (ν + 1)−1

for all ν ∈ N.

Proof.

ad 1. We adopt the theory and notations of [Hac85, Sec 7.1] and consider first the case ofa small overlap 0 < q ≤ q. We begin with the estimate of the ν-fold application ofthe smoothing operator, i.e., ‖Kν

k‖0←0.

Kνk = (Ink

− ωD−1k Lk)

ν = D−1/2k (Ink

− ωD−1/2k LkD

−1/2k )νD

1/2k .

Therefore,

‖Kνk‖0←0 ≤ ‖D−1/2

k ‖0←0‖(Ink− ωD

−1/2k LkD

−1/2k )ν‖0←0‖D1/2

k ‖0←0

≤ ‖(Ink− ωD

−1/2k LkD

−1/2k )ν‖0←0 .

By using the definitions of the single block matrices of Lk in (6.6) - (6.8) and LemmaA.3 we obtain

‖Lk‖0←0 ≤√

2(‖Ak‖2

0←0 + 2‖Bk‖20←0 + ‖Ck‖2

0←0

)1/2 ≤ CL = O(1) .

Hence, the assumption 0 < ω ≤ ω < C−1L < 1 leads to

‖Kνk‖0←0 ≤ 1

and the constant CS in (3.16) satisfies CS = 1. In order to estimate the constantsCp and Cp in (3.17) we have to investigate the eigenvalues of the symmetric matrix

p⊺

k,k−1pk,k−1 =(P−1k Pk−1

)⊺P−1k Pk−1 .

Remark 3.4 implies that the eigenvalues of p⊺

k,k−1pk,k−1 are bounded from below by1 and from above by 4. Hence,

‖uk−1‖0 ≤ ‖pk,k−1uk−1‖0 ≤ 4‖uk−1‖0

for all k ≥ 1 and C⋆ := 1 + CpCp (1 + CS) = O(1) (cf. [Hac85, (7.1.6)]). Let MGk

denote the iteration matrix of the W -cycle multigrid method and Tk the iterationmatrix of the twogrid method (cf. (5.1)). Then, according to [Hac85, (7.1.5c)] therecursion

‖MGk‖0←0 ≤ ‖Tk‖0←0 + C⋆‖MGk−1‖20←0


holds. Choose β in Theorem 6.1 as

β := (4C⋆Ca)−1

and the bounds ν, ν, ω, ω, q = O(1) accordingly. Then, [Hac85, Lemma 7.1.6]implies that

‖MGk‖0←0 ≤ 1

2C⋆≤ 1/2

yielding the multigrid convergence with respect to Euclidean norm.

ad 2. Let us now consider the case q ∈ [q, 1]. The norm equivalence

Cp‖uk−1‖0 ≤ ‖pk,k−1uk−1‖0 ≤ Cp‖uk−1‖0

does not depend on q. By using Remark 3.4 we obtain Cp = 1 and Cp = 4 also in thiscase. By using the second statement of Theorem 6.33, the standard multigrid theory,e.g., [Hac85, Theorem 7.1.2] with respect to the ‖·‖0-norm can be applied yieldingthe asserted estimate. The constant in the estimate only depends on q = O(1).

Summarising we have proved for a two-dimensional model problem that the multigridconvergence is robust with respect to a small overlap of elements with the domain.


7 Summary

In this thesis we have introduced a multigrid algorithm based on composite finite elementdiscretisations. We have analysed the robust convergence of this algorithm for a one- anda two-dimensional elliptic boundary value model problem. As a prototype of an ellipticboundary value problem of second order we have considered the Poisson equation onadequate one- and two-dimensional domains. These domains were chosen in such a way,that characteristic difficulties, as they arise in the solution procedure of boundary valueproblems on complicated domains, could be analysed. The multigrid method based oncomposite finite elements applied to a Neumann problem on the complicated domain ofthe Baltic sea converges robust.The presented multigrid algorithm is rather simple. The description of the algorithmdoes not depend on the complexity of the physical domain, i.e., on geometric details.The method is based on composite finite element discretisations of the boundary valueproblem. A composite finite element space is simply the restriction of a standard finiteelement space on an overlapping domain to the given geometry. We investigated theconvergence of the multigrid method following the general multigrid convergence theoryin [Hac85]. However, the proof requires a smoothing and an approximation propertywhich depends on the geometric details of the domain in a complicated way. We emphasisthat the formulation of the algorithm is very simple (no special smoothing or weightedprolongations etc. are employed). The convergence analysis is far from trivial and showsthe robust convergence with respect to geometrical details.

85

86 7 Summary

A Basics

This appendix comprehends general definitions and theorems, which are used in the thesis.For the proof of most of these theorems we refer to the standard textbook.

A.1 Linear Algebra

In a normed spaceX with norm ‖·‖X the open ball about the point x ∈ X with radius r > 0is denoted by BX

r (x) := y ∈ X : ‖x− y‖X < r and BXr (x) := y ∈ X : ‖x− y‖X ≤ r

denotes its closure.

Theorem A.1 (Gerschgorin) Let L = (Lij)ni,j=1 ∈ R

n×n be a real (n × n)-matrix andfor all 1 ≤ i ≤ n let

ri :=

n∑

j = 1j 6= i

|Lij | .

For all eigenvalues λ of L there holds

λ ∈n⋃

i=1

BCri(Lij) .

For the proof we refer to [Hac92, Criterion 4.3.4].The results of the convergence theory of the multigrid algorithm for a one- and a two-dimensional model problem presented in Chapter 5 and Chapter 6 are based on a parti-tioning of the system matrix L. Therefore, some properties of block matrices are recalledhere. Let L = (Lij)

ni,j=1 ∈ R

n×n be a real (n × n)-matrix of the form

L =:

[A BC D

]

, (A.1)

with A = (Aij)ℓi,j=1 ∈ R

ℓ×ℓ, B ∈ Rℓ×k, C ∈ R

k×ℓ and D = (Dij)ki,j=1 ∈ R

k×k.

Lemma A.2 Let ν > 0 be an integer. Let L ∈ Rn×n has a block partitioning as in (A.1)

with B = 0 ∈ Rℓ×k

L =:

[A 0C D

]

.

Then, the ν-th power of L can be written as

Lν =

[Aν 0

∑ν−1m=0 DmCAν−m−1 Dν

]

.

87

88 A Basics

Proof. We prove the lemma by induction. For ν = 1 and ν = 2 the assertion is satisfied,because

L =

[A 0

∑0m=0 DmCA−m D

]

and

L2 =

[A2 0

CA + DC D2

]

=

[A2 0

∑1m=0 DmCA1−m D2

]

.

By assuming that the assertion is satisfied for ν = i we may conclude the case ν = i + 1from

Li+1 =

[Ai 0

∑i−1m=0 DmCAi−m−1 Di

] [A 0C D

]

=

[Ai+1 0

∑i−1m=0 DmCAi−m + DiC Di+1

]

=

[Ai+1 0

∑im=0 DmCAi−m Di+1

]

For the proof of the smoothing and of the approximation property norm estimates of blockstructured matrices are employed.

Lemma A.3 Let L ∈ R(n+m)×(n+m) be a matrix with blockstructure, given by

L =

[A BC D

]

with A ∈ Rn×n, B ∈ R

n×m, C ∈ Rm×n and D ∈ R

m×m. Then,

‖L‖0←0 ≤√

2(‖A‖2

0←0 + ‖B‖20←0 + ‖C‖2

0←0 + ‖D‖20←0

).

Proof. Let v ∈ Rn+m be partitioned according to v = [v1,v2]

⊺ with v1 ∈ Rn and

v2 ∈ Rm.

‖L‖20←0 = sup

v∈Rn+m\0

‖Lv‖20

‖v‖20

with ‖v‖20 =

n+m∑

i=1

v2i .

Without lost of generality we assume n ≥ m. Explicit calculation yield

‖Lv‖20←0 =

∥∥∥∥

[Av1 + Bv2

Cv1 + Dv2

]∥∥∥∥

2

0

=

n∑

i=1

(Av1 + Bv2)2i +

m∑

i=1

(Cv1 + Dv2)2i .

Using the binomial inequality

(a+ b)2 = a2 + 2ab+ b2 ≤ 2(a2 + b2)

it follows

‖Lv‖20←0 ≤ 2

(n∑

i=1

((Av1)

2i + (Bv2)

2i

)+

m∑

i=1

((Cv1)

2i + (Dv2)

2i

)

)

= 2(‖Av1‖2

0 + ‖Bv2‖20 + ‖Cv1‖2

0 + ‖Dv2‖20

).

A.2 Hilbert Spaces 89

Therefore,

‖L‖20←0 ≤ 2 sup

v = (v1,v2)⊺

v1 ∈ Rn \ 0v2 ∈ Rm \ 0

(‖Av1‖20

‖v‖20

+‖Bv2‖2

0

‖v‖20

+‖Cv1‖2

0

‖v‖20

+‖Dv2‖2

0

‖v‖20

)

≤ 2 supv1∈Rn\0

(‖Av1‖20

‖v1‖20

+‖Cv1‖2

0

‖v1‖20

)

+ 2 supv2∈Rm\0

(‖Bv2‖20

‖v2‖20

+‖Dv2‖2

0

‖v2‖20

)

≤ 2(‖A‖2

0←0 + ‖B‖20←0 + ‖C‖2

0←0 + ‖D‖20←0

).

A.2 Hilbert Spaces

Theorem A.4 (Riesz Representation Theorem) Let (H, (·, ·)) a real Hilbert spaceand let J : H → H ′ be defined by

J(y)(x) := (x, y) ∀x, y ∈ H .

Then J is linear, isometric and bijective.

For the proof we refer to [Alt85, Satz 4.6].

Definition A.5 Let V be a Hilbert space.

1. The mapping a(·, ·) : V × V → R is called a bilinear form if

a(x, y + λz) = a(x, y) + λa(x, z) , a(x+ λy, z) = a(x, z) + λa(y, z)

for all x, y, z ∈ V and λ ∈ R.

2. a(·, ·) is said to be continuous (or bounded) if there exists a CΩ such that

|a(x, y)| ≤ CΩ‖x‖V ‖y‖V ∀x, y ∈ V .

3. a(·, ·) is said to be coercive if there exists a Ccoerc such that

a(x, x) ≥ Ccoerc‖u‖2V ∀x ∈ V .

Theorem A.4 is applied in the proof of the following theorem.

Theorem A.6 (Lax-Milgram Theorem, first version) Let (H, (·, ·)) a real Hilbertspace and a : H × H → R a bilinear form with the properties: There exist constants0 < c ≤ C <∞ such that

|a(x, y)| ≤ C ‖x‖‖y‖ (Continuity),

a(x, x) ≥ c ‖x‖2 (Coercivity)

for all x, y ∈ H. Then there exists one and only one linear and bijective mapping A : H →H such that

a(x, y) = (x,Ay) ∀x, y ∈ H .

90 A Basics

For the proof we again refer to [Alt85, Satz 4.7].

Lemma A.7 (Lax-Milgram Theorem, second version) Let (H, (·, ·)) a real Hilbertspace, a : H ×H → R a continuous and coercive bilinear form and f ∈ H ′. Then thereexists a unique y ∈ H such that

a(x, y) = f(x) ∀x ∈ H .

Proof. From the Riesz representation theorem A.4, it follows that there exists a uniquez ∈ H with f = (·, z). With the assertion of the Lax-Milgram theorem A.6 there exists alinear and bijective mapping A : H → H with a(·, ·) = (·, A·). Set y := A−1x, then for allx ∈ H it holds

a(x, y) = (x,Ay) = (x, x) = f(x) .

Theorem A.8 (Aubin-Nitsche-Lemma) Let H be a real Hilbert space with the norm‖ · ‖H and the scalar product (·, ·)H . Suppose V is a subspace of H, which becomes withthe scalar product (·, ·)H a Hilbert space. Furthermore, let u ∈ V be the exact solution ofthe variational problem: Find u ∈ V such that

a(u, v) =

∫

Ω

fv d x ∀v ∈ V .

Let V → H be continuous. Then for the finite element solution uS ∈ S ⊂ V there holds

‖u− us‖V ≤ C‖u− uS‖H supg∈H\0

1

‖g‖Vinfv∈S

‖ϕg − v‖H

,

if for any g ∈ H there exists a unique weak solution ϕg ∈ V of the variational problem

a(w,ϕg) =

∫

Ω

gw dx ∀w ∈ V .

For the proof we refer to [Bra97, Lemma 7.6]

A.3 Domains and Grids

A domain Ω ⊂ Rd is an open and connected set. A d-simplex with vertices xi ∈ R

d,i = 1, . . . , n, is given by

int (conv xi : i = 1, . . . n) ,

that means, by the inside of the convex closure of the vertices.

Definition A.9 (Partitioning) 1. Let Ω ⊂ R be a bounded domain, i.e., an openand bounded interval. Let I ⊂ N be finite and let τii∈I be a family of open anddisjoint intervals with the property

Ω =⋃

i∈I

τi .

A partitioning T of the domain Ω is given by τii∈I .

A.3 Domains and Grids 91

2. Let Ω ⊂ R2 be a bounded domain with polygonal boundary. Let I ⊂ N be finite

and let τii∈I be a family of open and disjoint simplices with the property

Ω =⋃

i∈I

τi .

The finite collection T := τii∈I is denoted a partitioning of Ω.

Definition A.10 (Grid, Triangulation) A triangulation G of a polygonal domain Ω isa partitioning consisting of triangles τi, i ∈ I having the property that for i 6= j thereholds either

1. τ i ∩ τ j = ∅ , or

2. τ i ∩ τ j is a common vertex of the elements τi and τj, or

3. τ i ∩ τ j is a common edge of τi and τj.

Θ denotes the set of all vertices of triangles τi ∈ G.

Definition A.11 Let Ω be a given domain and let T h, 0 < h ≤ 1, be a family oftriangulations. Let hτ be the longest side of τi ∈ T h, then h := maxhτ : τi ∈ T h is thelongest side-length in T h. Denote by ρτ the radius of the inscribed circle in τi ∈ T h.

1. The family is said to be quasi-uniform if there exists ρ > 0 such that

maxhτ/ρτ : τi ∈ T h ≥ ρ ∀h ∈ (0, 1] .

2. The family is said to be shape-regular if there exists ρ > 0 such that for all τi ∈ T h

h ≥ ρ minρτ : τi ∈ T h

and for which h→ 0 as the number of triangles ♯I tends to infinity.

Definition A.12 (Lipschitz boundary) 1. The domain Ω ⊂ Rd is called a special

Lipschitz domain, if there exists a Lipschitz-continuous function g : Rd−1 → R and

an orthogonal transformation Φ : Rd → R

d with the property

Ω = Φ(

(x, y) ∈ Rd−1 × R : y > g(x)

)

.

2. Let Ω ⊂ Rd be open. The boundary ∂Ω of Ω is called a Lipschitz boundary, if for

all x ∈ ∂Ω there exist a neighbourhood Ux of x and a special Lipschitz domain Ωx

such that

Ux ∩ Ω = Ux ∩ Ωx .

Remark A.13 If Ω in Definition A.12 (2.) is bounded, then ∂Ω is compact and thefamily (Ux)x∈∂Ω of ∂Ω has a finite subfamily.

92 A Basics

A.4 Functional Spaces

In the sequel all functions are assumed to be real-valued.

Definition A.14 Let Ω ⊂ Rd be open and measurable. C∞(Ω) denotes the space of

infinitely continuously differentiable functions on Ω. Let

C∞(Ω) :=

f |Ω : f ∈ C∞(Rd)

and

C∞0 (Ω) :=

f ∈ C∞(Rd) : supp f ⊂ Ω is compact

.

Let k ∈ N0. Let Pk = Pd,k be the space of all polynomials in d variables with degree lessthan or equal to k.

M(Ω,R) denotes the set of all measurable functions with real values. By identifying twofunctions if they are equal almost everywhere, an equivalence relation on M(Ω,R) is given.Consider the set of all measurable functions from Ω to R whose absolute value raised tothe p-th power has a finite Lebesgue integral, i.e.,

‖f‖p :=

∫

Ω

|f(x)|p dx

1/p

<∞ .

This set of equivalence classes together with the function ‖ · ‖p is the Banach space Lp(Ω)or more precisely Lp(Ω,Rd). The space of square-integrable functions L2(Ω) is a Hilbertspace with scalar product

(f, g)L2(Ω) :=

∫

Ω

fg d x ∀f, g ∈ L2(Ω) .

Let L1loc(Ω) denote the space of all locally integrable functions.

Definition A.15 ∂j denotes the partial derivation in the direction of the j-th unit vector.For t ∈ N

d0 let

Dt :=d∏

j=1

∂tjj .

Let f ∈ L1loc(Ω). The t-th weak derivation of f exists, if there exists gt ∈ L1

loc(Ω) suchthat for all φ ∈ C∞0 (Ω)

∫

fDtφ dx = (−1)|t|∫

Ω

gtφ dx .

Then the t-th weak derivation of f is given by Dtf := gt.

For k ∈ N

W k,p(Ω) :=

f ∈ Lp(Ω) : ∀t ∈ Nd0 with |t| ≤ k is Dtf ∈ Lp(Ω)

A.5 Theorems for Sobolev Spaces 93

is the Sobolev space W k,p(Ω) and Hk(Ω) := W k,2(Ω). Note that H0(Ω) coincides withL2(Ω). In W k,p(Ω0 we define the norm ‖ · ‖W k,p(Ω) and the semi-norm | · |W k,p(Ω) by

‖f‖W k,p(Ω) :=

∑

|t|≤k

‖Dtf‖pLp(Ω)

1/p

and |f |W k,p(Ω) :=

∑

|t|=k

‖Dtf‖pLp(Ω)

1/p

and in Hk(Ω) the scalar product

(f, g)Hk(Ω) :=∑

|t|≤k

∫

DtfDtg d x .

Finally let Hk0 (Ω) be the closure of C∞0 (Ω) with respect to the Hk-norm, i.e.,

Hk0 (Ω) := C∞0 (Ω)

‖·‖Hk(Ω) .

Theorem A.16 The space W k,p(Ω) is a Banach space.

For the proof we refer to [Ada75, Theorem 3.2].

Remark A.17 Since the space Hk0 (Ω) is closed in Hk(Ω) it is a Banach space.

A.5 Theorems for Sobolev Spaces

Theorem A.18 (Poincare-Friedrichs-Inequality) Let Ω ⊂ Rd an open set which is

contained in a cuboid with length of the edge s ∈ (0,∞). For all u ∈ H10 (Ω) there holds

‖u‖H1(Ω) ≤ (1 + s)|u|H1(Ω) .

For the proof we refer to [Bra97, Chapter 2, Theorem 1.5].

Theorem A.19 Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary. Then the

space C∞(Ω) is dense in W k,p(Ω).

The proof is given in [Maz85, Theorem 1.1.6/2].

Lemma A.20 Let V be a Hilbert space and let V1 and V2 be dense in V . Let a(·, ·) bea continuous bilinear form defined on V1 × V2. Then a(·, ·) can be extended uniquely toV × V so that the continuity condition holds with the same constant c for all x, y ∈ V .

For the proof we refer, e.g., to [Hac92, Lemma 6.5.1.].

Theorem A.21 (Green’s Formula) Let Ω ⊂ Rd be a bounded domain with Lipschitz

boundary. Let u ∈ H2(Ω) and v ∈ H1(Ω) and let ν denote the outward unit normal vectorto ∂Ω, which is by assumption in L∞(∂Ω)d. Then

∫

Ω

(−∆u)v d x =

∫

Ω

∇u · ∇v d x−∫

∂Ω

∂u

∂νv .

94 A Basics

For the proof we refer, e.g., to [BS94, Proposition (5.1.6)].The following theorem is a part of Sobolev’s Inequality, see e.g. [BS94, Theroem (1.4.6)].

Theorem A.22 (Sobolev’s Inequality) Let Ω ⊂ Rd be a bounded domain with Lip-

schitz boundary. Let p ∈ [1,∞), and let k and m be positive integers satisfying m < k.

1. If

k −m ≥ d for p = 1 ,

k −m > d/p for p > 1

then there exists a constant C > 0 such that for all u ∈W k.p(Ω) it is

‖u‖Wm,∞(Ω) ≤ C‖u‖W k,p(Ω) .

Furthermore u has a representative in Cm(Ω).

2. If d < kp then, for each function in W k,p(Ω), there exists a representative in C(Ω).

Proof.

ad 1. The proof of the first part of Theorem A.22 is given in [BS94, Corollary (1.4.7)].

ad 2. Let u ∈ W k,p(Ω) and l ∈ N minimal with d < lp. Then u ∈ W l,p(Ω) and with[Ada75, Lemma 5.17] u has a representative in C(Ω).

Lemma A.23 Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary and suppose

that p is a real number in the range of 1 ≤ p ≤ ∞. Then there exists one and only onebounded trace operator Ψ : W 1,p(Ω) → Lp(Ω) with the property

Ψu = u|∂Ω ∀u ∈W 1,p(Ω) ∩C(Ω) .

For the proof we refer to [Alt85, A 6.6].

Theorem A.24 Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary and suppose

that p is a real number in the range of 1 ≤ p ≤ ∞. Then there is a constant C such that

‖v‖Lp(∂Ω) ≤ C‖v‖1−1/pLp(Ω)‖v‖

1/pW 1,p(Ω)

∀v ∈W 1,p(Ω) .

This assertion is a direct conclusion of Lemma A.23 and the proof is given in [BS94,Theorem (1.6.6)].

B Numerical Tests

Numerical tests for the estimate of ‖VII‖0←0 and of ‖VIII‖0←0 in the proof of theapproximation property for the two-dimensional model problem, see Section 6.3:

We first estimate the norm of YIIRCk−1L

−1k−1 (see Table B.1) with

YII = −qkQRRk pCC

k,k−1 + qk−1pRRk,k−1Qk−1 .

kq−1k ‖YIIR

Ck−1L

−1k−1‖0←0 1 2 3 4 5

2−2 0.179 0.211 0.218 0.220 0.2202−4 0.253 0.299 0.309 0.311 0.3122−6 0.280 0.331 0.341 0.344 0.3442−8 0.288 0.340 0.350 0.352 0.353

qk 2−10 0.290 0.342 0.352 0.355 0.3562−12 0.290 0.342 0.353 0.355 0.3562−14 0.290 0.343 0.353 0.355 0.3562−16 0.290 0.343 0.353 0.355 0.356

Table B.1: ‖YIIRCk−1L

−1k−1‖0 is bounded by Cqk.

By using a triangle inequality we obtain

‖YIIRCk−1L

−1k−1‖0←0 ≤ ‖qkQRR

k pCCk,k−1R

Ck−1L

−1k−1‖0←0+‖qk−1p

RRk,k−1Q

RRk−1R

Ck−1L

−1k−1‖0←0 .

In Table B.2 and Table B.3 we investigate the two summands of the right hand side aboveseparately.We split the matrix YIIR

Ck−1L

−1k−1 into the factors YII and RC

k−1L−1k−1 and estimate in

Table B.4 ‖RCk−1L

−1k−1‖0←0.

Now, we analyse the influence of the eigenvalues of QRRk on this norm, see Table B.5.

Let gk ∈ Rnk with (gk)x = 0 for all x ∈ ΘR

k .Applying the Schur-complement representationfor the inverse of Lk yields

QRRk RC

kL−1k gk = QRR

k RLCk A−1

k RLkgk + QRR

k RLCk A−1

k B⊺

kS−1k BkA

−1k RL

kgk .

For the Euclidean norm of the matrix QRRk RLC

k A−1k we get the following numerical results:

‖QRRk RLC

k A−1k ‖0←0 ≤ C but ‖RLC

k A−1k ‖0←0 ≤ Ch

−3/2k ,

see Table B.6 and Table B.7.

95

96 B Numerical Tests

k‖QRRk pCC

k,k−1RCk−1L

−1k−1‖0←0 1 2 3 4 5

2−2 0.281 0.307 0.310 0.311 0.3112−4 0.368 0.427 0.438 0.440 0.4412−6 0.399 0.469 0.483 0.486 0.4872−8 0.407 0.480 0.495 0.498 0.499

qk 2−10 0.410 0.483 0.498 0.502 0.5032−12 0.410 0.484 0.499 0.502 0.5032−14 0.410 0.484 0.499 0.503 0.5032−16 0.410 0.484 0.499 0.503 0.504

Table B.2: ‖qkQRRk pCC

k,k−1RCk−1L

−1k−1‖0←0 is bounded by Cqk.

k‖pRRk,k−1Q

RRk−1R

Ck−1L

−1k−1‖0←0 1 2 3 4 5

2−2 0.428 0.444 0.444 0.444 0.4442−4 0.535 0.609 0.620 0.623 0.6242−6 0.569 0.665 0.683 0.687 0.6882−8 0.577 0.680 0.700 0.705 0.706

qk 2−10 0.580 0.684 0.704 0.709 0.7122−12 0.580 0.685 0.705 0.710 0.7122−14 0.580 0.685 0.706 0.711 0.7122−16 0.580 0.685 0.706 0.711 0.712

Table B.3: ‖qk−1pRRk,k−1Q

RRk−1R

Ck−1L

−1k−1‖0←0 is bounded by Cqk.

kh

3/2k ‖RC

k L−1k ‖0←0 1 2 3 4 5

2−2 0.189 0.169 0.151 0.140 0.1332−4 0.217 0.182 0.157 0.142 0.1342−6 0.226 0.185 0.158 0.143 0.1342−8 0.228 0.186 0.159 0.143 0.134

qk 2−10 0.229 0.186 0.159 0.143 0.1342−12 0.229 0.186 0.159 0.143 0.1342−14 0.229 0.186 0.159 0.143 0.1342−16 0.229 0.186 0.159 0.143 0.134

Table B.4: ‖RCk L−1

k ‖0←0 is bounded by Ch−3/2k .

97

k‖QRRk RC

k L−1k ‖0←0 1 2 3 4 5

2−2 0.472 0.472 0.473 0.474 0.4742−4 0.770 0.778 0.779 0.779 0.7792−6 0.912 0.935 0.940 0.941 0.9422−8 0.954 0.982 0.988 0.990 0.990

qk 2−10 0.965 0.994 1.001 1.003 1.0062−12 0.968 0.997 1.004 1.006 1.0062−14 0.969 0.998 1.005 1.007 1.0072−16 0.969 0.998 1.005 1.007 1.007

Table B.5: ‖QRRk RC

k L−1k ‖0←0 is bounded by a constant C.

k‖QRRk RLC

k A−1k ‖0←0 1 2 3 4 5

2−2 0.347 0.349 0.349 0.349 0.3492−4 0.564 0.573 0.575 0.576 0.5762−6 0.656 0.672 0.676 0.677 0.6772−8 0.682 0.701 0.706 0.707 0.707

qk 2−10 0.689 0.709 0.714 0.715 0.7172−12 0.691 0.711 0.716 0.717 0.7172−14 0.691 0.711 0.716 0.717 0.7182−16 0.691 0.711 0.716 0.717 0.718

Table B.6: ‖QRRk RLC

k A−1k ‖0←0 is bounded by C.

kh

3/2k ‖RLC

k A−1k ‖0←0 1 2 3 4 5

2−2 0.117 0.088 0.060 0.037 0.0212−4 0.153 0.127 0.104 0.080 0.0572−6 0.166 0.143 0.127 0.113 0.0972−8 0.170 0.148 0.135 0.127 0.119

qk 2−10 0.171 0.149 0.137 0.130 0.1282−12 0.171 0.150 0.138 0.131 0.1282−14 0.171 0.150 0.138 0.131 0.1282−16 0.171 0.150 0.138 0.132 0.128

Table B.7: ‖RLCk A−1

k ‖0←0 is bounded by Ch−3/2k .

98 B Numerical Tests

C Notation

a(·, ·) bilinear formC generic constantd dimension of Ω

Dk diagonal part of the system matrix Lk∆ Laplace operator ∆ :=

∑di=1 ∂

2/∂x2i

Γ boundary of ΩΓD Dirichlet boundaryΓN Neumann boundaryhk discretisation parameter (grid size) at level k

Hm(Ω) Sobolev spacesHm

0 (Ω) closure of C∞0 (Ω) with respect to ‖ · ‖Hm(Ω)

H1D(Ω) energy space H1

D(Ω) := u ∈ H1(Ω) : u|ΓD= 0

I interpolation on Sn(Ω)In identity matrix of dimension nk discretisation level numberL operator associated with the boundary value problem

Lk system matrix on discretisation level kL2(Ω) space of square integrable functions over Ω

Mk mass matrix on discretisation level kMGk multigrid iteration matrix

nk number of variables on level kν1 number of pre-smoothing stepsν2 number of post-smoothing steps0 zero matrix

O(·), o(·) Landau symbolsΩ bounded domain in R

d

ω relaxation factor of the damped Jacobi iterationPk isomorphism from R

nk onto SCFEk (Ω)

pk,k−1 prolongation operatorRk the adjoint of Pk, 〈Rku,v〉 = (u, Pkv)L2(Ω)

rk−1,k restriction operator(A) spectral radius of matrix A

Sk(Ω) = SCFEk (Ω) composite finite element space on ΩSk(Ωk) standard finite element space on Ωk

Sn(Ω) = S(Ω) standard finite element space on Ωσ(A) spectrum of matrix A

Tk twogrid iteration matrix with respect to the levels k and k − 1uk finite element solution on level kuS discrete solution

99

100 C Notation

〈·, ·〉 Euclidean scalar product(·, ·)L2(Ω) L2 scalar product of Hm(Ω)

| · | real absolute value‖ · ‖0 Euclidean norm

‖ · ‖0,Ω = ‖ · ‖0 discrete L2-norm on Sk(Ω)

‖ · ‖Hm(Ω) Sobolev norm on Hm(Ω)

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Lebenslauf

von Nadin Stahn, geborene Fraubose

Geburtstag: 09. 04. 1975Geburtsort: Brandenburg a.d. Havel, DeutschlandFamilienstand: verheiratet mit Dr. Jochen Stahn seit 07.02.2003Kinder: Leonhard Stahn, geboren am 25.05.2003Staatsangehorigkeit: deutsch

Schulischer und wissenschaftlicher Werdegang

09. 1981 – 07. 1990 Besuch der allgemeinbildenden Schule Franz-Ziegler,Brandenburg a.d. Havel;

09. 1990 – 07. 1993 Besuch des Gymnasiums Nord, Brandenburg a.d. Havel,Abschluss mit allgemeiner Hochschulreife;

10. 1993 – 02. 1999 Mathematikstudium an der Universitat Potsdam;25. 05. 1998 Vordiplomsprufung im Hauptfach Mathematik und im

Nebenfach Physik;03. 1999 – 01. 2000 Diplomarbeit bei Prof. Dr. Joachim Grater,

Universitat Potsdam, zum Thema Uber die Faktorisierungganzer Zahlen mit elliptischen Kurven;

08. 05. 2000 Diplomprufung Mathematik;

seit 08. 2000 Doktorat an der Universitat Zurich, Anfertigung derDoktorarbeit Composite Finite Elements and Multigridunter der Leitung von bei Prof. Dr. Stefan Sauter,Numerische Analysis, Institut fur Mathematik.

Nadin Stahn Zurich, 19. April 2006

Danksagung

Diese Arbeit soll dokumentieren, dass ich selbstandig wissenschaftlich arbeiten kann. Den-noch war ihr Zustandekommen in dieser Form nur moglich durch das Wirken vieler Mit-menschen, sei es durch direkte Mitarbeit, durch Diskussionen oder einfach dadurch, dasssie mir eine schone Zeit in Zurich ermoglicht haben.

Zu allererst danke ich Herrn Prof.Dr. Sauter fur die Betreuung dieser Arbeit. Insbeson-dere bedanke ich mich dafur, dass ich so viele Konferenzen und Seminare besuchen konnte:Ich habe davon sehr profitiert!

Bei Dirk Feuchter und Ingo Heppner bedanke ich mich fur die Unterstutzung bei den nu-merischen Tests.

Eine ganze Reihe von Anregungen, die hier ihren Niederschlag gefunden haben, verdankeich Gesprachen mit Rainer, Nico, Steffen, Carsten, Lars und vielen anderen. Dass wirauch uber andere Themen gesprochen haben, hat das Arbeiten erst angenehm gemacht.

Viel Unterstutzung in nicht-mathematischen Gesprachen habe ich auch von Bernadette,Nadja und Lili erhalten.

Meinen Eltern danke ich dafur, dass sie mich immer darin bestarkt haben, dass ich dasRichtige tue.

Leonhard: Danke, dass Mama so viele Einsen malen durfte!

Jochen: Dir kann ich nicht mit Worten danken!

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