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Adaptive Finite Element MethodsLecture Notes Winter Term 2011/12

R. Verfurth

Fakultat fur Mathematik, Ruhr-Universitat Bochum


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Contents

Chapter I. Introduction 7I.1. Motivation 7I.2. One-dimensional Sobolev and finite element spaces 11I.2.1. Sturm-Liouville problems 11

I.2.2. Idea of the variational formulation 11I.2.3. Weak derivatives 12I.2.4. Sobolev spaces and norms 12I.2.5. Variational formulation of the Sturm-Liouville problem 13I.2.6. Finite element spaces 14I.2.7. Finite element discretization of the Sturm-Liouville

problem 14I.2.8. Nodal basis functions 15I.3. Multi-dimensional Sobolev and finite element spaces 15I.3.1. Domains and functions 15I.3.2. Differentiation of products 16

I.3.3. Integration by parts formulae 16I.3.4. Weak derivatives 16I.3.5. Sobolev spaces and norms 17I.3.6. Friedrichs and Poincare inequalities 18I.3.7. Finite element partitions 19I.3.8. Finite element spaces 21I.3.9. Approximation properties 22I.3.10. Nodal shape functions 23I.3.11. A quasi-interpolation operator 25I.3.12. Bubble functions 26

Chapter II. A posteriori error estimates 29II.1. A residual error estimator for the model problem 30II.1.1. The model problem 30II.1.2. Variational formulation 30II.1.3. Finite element discretization 30II.1.4. Equivalence of error and residual 30II.1.5. Galerkin orthogonality 31II.1.6. L2-representation of the residual 32II.1.7. Upper error bound 33II.1.8. Lower error bound 35

II.1.9. Residual a posteriori error estimate 383


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4 CONTENTS

II.2. A catalogue of error estimators for the model problem 40II.2.1. Solution of auxiliary local discrete problems 40

II.2.2. Hierarchical error estimates 46II.2.3. Averaging techniques 51II.2.4. Equilibrated residuals 53II.2.5. H(div)-lifting 57II.2.6. Asymptotic exactness 60II.2.7. Convergence 62II.3. Elliptic problems 62II.3.1. Scalar linear elliptic equations 62II.3.2. Mixed formulation of the Poisson equation 65II.3.3. Displacement form of the equations of linearized

elasticity 68II.3.4. Mixed formulation of the equations of linearizedelasticity 70

II.3.5. Non-linear problems 76II.4. Parabolic problems 77II.4.1. Scalar linear parabolic equations 77II.4.2. Variational formulation 79II.4.3. An overview of discretization methods for parabolic

equations 79II.4.4. Space-time finite elements 80II.4.5. Finite element discretization 81

II.4.6. A preliminary residual error estimator 82II.4.7. A residual error estimator for the case of small

convection 84II.4.8. A residual error estimator for the case of large

convection 84II.4.9. Space-time adaptivity 85II.4.10. The method of characteristics 87II.4.11. Finite volume methods 89II.4.12. Discontinuous Galerkin methods 95

Chapter III. Implementation 97III.1. Mesh-refinement techniques 97III.1.1. Marking strategies 97III.1.2. Regular refinement 100III.1.3. Additional refinement 101III.1.4. Marked edge bisection 103III.1.5. Mesh-coarsening 103III.1.6. Mesh-smoothing 105III.2. Data structures 108III.2.1. Nodes 109III.2.2. Elements 109

III.2.3. Grid hierarchy 110


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CONTENTS 5

III.3. Numerical examples 110

Chapter IV. Solution of the discrete problems 119IV.1. Overview 119IV.2. Classical iterative solvers 122IV.3. Conjugate gradient algorithms 123IV.3.1. The conjugate gradient algorithm 123IV.3.2. The preconditioned conjugate gradient algorithm 126IV.3.3. Non-symmetric and indefinite problems 129IV.4. Multigrid algorithms 131IV.4.1. The multigrid algorithm 131IV.4.2. Smoothing 134IV.4.3. Prolongation 135

IV.4.4. Restriction 136

Bibliography 139

Index 141


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CHAPTER I

Introduction

I.1. Motivation

In the numerical solution of practical problems of physics or engi-neering such as, e.g., computational fluid dynamics, elasticity, or semi-conductor device simulation one often encounters the difficulty that

the overall accuracy of the numerical approximation is deteriorated bylocal singularities such as, e.g., singularities arising from re-entrant cor-ners, interior or boundary layers, or sharp shock-like fronts. An obviousremedy is to refine the discretization near the critical regions, i.e., toplace more grid-points where the solution is less regular. The questionthen is how to identify those regions and how to obtain a good bal-ance between the refined and un-refined regions such that the overallaccuracy is optimal.

Another closely related problem is to obtain reliable estimates of theaccuracy of the computed numerical solution. A priori error estimates,as provided, e.g., by the standard error analysis for finite element orfinite difference methods, are often insufficient since they only yieldinformation on the asymptotic error behaviour and require regularityconditions of the solution which are not satisfied in the presence ofsingularities as described above.

These considerations clearly show the need for an error estimatorwhich can a posteriori be extracted from the computed numerical so-lution and the given data of the problem. Of course, the calculationof the a posteriori error estimate should be far less expensive than thecomputation of the numerical solution. Moreover, the error estimatorshould be local and should yield reliable upper and lower bounds for

the true error in a user-specified norm. In this context one should note,that global upper bounds are sufficient to obtain a numerical solutionwith an accuracy below a prescribed tolerance. Local lower bounds,however, are necessary to ensure that the grid is correctly refined sothat one obtains a numerical solution with a prescribed tolerance usinga (nearly) minimal number of grid-points.

Disposing of an a posteriori error estimator, an adaptive mesh-refinement process has the following general structure:

Algorithm I.1.1. (General adaptive algorithm)

(0) Given: The data of a partial differential equation and a toler-ance .

7


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8 I. INTRODUCTION

Sought: A numerical solution with an error less than .(1) Construct an initial coarse mesh

T0 representing sufficiently

well the geometry and data of the problem. Set k = 0.(2) Solve the discrete problem on Tk.(3) For each element K in Tk compute an a posteriori error esti-

mate.(4) If the estimated global error is less than then stop. Otherwise

decide which elements have to be refined and construct the nextmesh Tk+1. Replace k by k + 1 and return to step (2).

The above algorithm is best suited for stationary problems. Fortransient calculations, some changes have to be made:

The accuracy of the computed numerical solution has to beestimated every few time-steps.

The refinement process in space should be coupled with a time-step control.

A partial coarsening of the mesh might be necessary. Occasionally, a complete re-meshing could be desirable.

In both stationary and transient problems, the refinement and un-refinement process may also be coupled with or replaced by a moving-point technique, which keeps the number of grid-points constant butchanges there relative location.

In order to make Algorithm I.1.1 operative we must specify

a discretization method, a solver for the discrete problems, an error estimator which furnishes the a posteriori error esti-

mate, a refinement strategy which determines which elements have

to be refined or coarsened and how this has to be done.

The first point is a standard one and is not the objective of these lecturenotes. The second point will be addressed in chapter IV (p. 119). Thethird point is the objective of chapter II (p. 29). The last point willbe addressed in chapter III (p. 97).

In order to get a first impression of the capabilities of such an adap-tive refinement strategy, we consider a simple, but typical example. Weare looking for a function u which is harmonic, i.e. satisfies

u = 0,in the interior of a circular segment centered at the origin with radius1 and angle 3

2, which vanishes on the straight parts D of the boundary

, and which has normal derivative 23

sin( 23

) on the curved part Nof . Using polar co-ordinates, one easily checks that

u = r2/3

sin(

2

3).


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I.1. MOTIVATION 9

Figure I.1.1. Triangulation obtained by uniform refinement

We compute the Ritz projections uT ofu onto the spaces of continuouspiecewise linear finite elements corresponding to the two triangulationsshown in figures I.1.1 and I.1.2, i.e., solve the problem:

Find a continuous piecewise linear function uT such that

uT vT =

N

2

3sin(

2

3)vT

holds for all continuous piecewise linear functions vT.The triangulation of figure I.1.1 is obtained by five uniform refinementsof an initial triangulation T0 which consists of three right-angled isosce-les triangles with short sides of unit length. In each refinement stepevery triangle is cut into four new ones by connecting the midpointsof its edges. Moreover, the midpoint of an edge having its two end-points on is projected onto . The triangulation in figure I.1.2is obtained from T0 by applying six steps of the adaptive refinementstrategy described above using the error estimator R,K of section II.1.9(p. 38). A triangle K Tk is divided into four new ones if

R,K 0.5 maxKTk

R,K

(cf. algorithm III.1.1 (p. 97)). Midpoints of edges having their twoendpoints on are again projected onto . For both meshes we listin table I.1.1 the number NT of triangles, the number NN of unknowns,and the relative error

=(u uT)

u.


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10 I. INTRODUCTION

It clearly shows the advantages of the adaptive refinement strategy.

Table I.1.1. Number of triangles NT and of unknownsNN and relative error for uniform and adaptive refine-ment

refinement NT NN uniform 6144 2945 0.8%adaptive 3296 1597 0.9%

Figure I.1.2. Triangulation obtained by adaptive refinement

Some of the methods which are presented in these lecture notes aredemonstrated in the Java applet ALF (Adaptive Linear Finite elements).It is available under the address

http://www.rub.de/num1/files/ALF.html.

A user guide can be found in pdf-form athttp://www.rub.de/num1/files/ALFUserGuide.pdf .

ALF in particular offers the following options:

various domains including those with curved boundaries, various coarsest meshes, various differential equations, in particular

the Poisson equation with smooth and singular solutions, reaction-diffusion equations in particular those having so-

lutions with interior layers, convection-diffusion equations in particular those with so-

lutions having interior and boundary layers,
http://www.rub.de/num1/files/ALF.htmlhttp://www.rub.de/num1/files/ALFUserGuide.pdfhttp://www.rub.de/num1/files/ALFUserGuide.pdfhttp://www.rub.de/num1/files/ALF.html


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I.2. ONE-DIMENSIONAL SOBOLEV AND FINITE ELEMENT SPACES 11

various options for building the stiffness matrix and the right-hand side,

various solvers in particular CG- and PCG-algorithms, several variants of multigrid algorithms with various cy-

cles and smoothers, the option to choose among uniform and and adaptive refine-

ment based on various a posteriori error estimators.

I.2. One-dimensional Sobolev and finite element spaces

I.2.1. Sturm-Liouville problems. Variational formulations andassociated Sobolev and finite element spaces are fundamental concepts

throughout these notes. To gain a better understanding of these con-cepts we first briefly consider the following one-dimensional boundaryvalue problem (Sturm-Liouville problem)

(pu) + qu = f in (0, 1)u(0) = 0, u(1) = 0.

Here p is a continuously differentiable function with

p(x) p > 0

for all x (0, 1) and q is a non-negative continuous function.I.2.2. Idea of the variational formulation. The basic idea of

the variational formulation of the above Sturm-Liouville problem canbe described as follows:

Multiply the differential equation with a continuously differ-entiable function v with v(0) = 0 and v(1) = 0:

(pu)(x)v(x) + q(x)u(x)v(x) = f(x)v(x)for 0 x 1.

Integrate the result from 0 to 1:1

0

(pu)(x)v(x) + q(x)u(x)v(x)dx = 10

f(x)v(x)dx.

Use integration by parts for the term containing derivatives:

1

0

(pu)(x)v(x)dx

= p(0)u(0) v(0)=0

p(1)u(1) v(1)=0

+

10

p(x)u(x)v(x)dx

= 1

0

p(x)u(x)v(x)dx.


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12 I. INTRODUCTION

To put these ideas on a profound basis we must better specify the prop-erties of the functions u and v. Classical properties such as continuous

differentiability are too restrictive; the notion derivative must be gen-eralised in a suitable way. In view of the discretization the new notionshould in particular cover piecewise differentiable functions.

I.2.3. Weak derivatives. The above considerations lead to thenotion of a weak derivative. It is motivated by the following obser-vation: Integration by parts yields for all continuously differentiablefunctions u and v satisfying v(0) = 0 and v(1) = 01

0

u(x)v(x)dx = u(1) v(1)

=0u(0) v(0)

=01

0

u(x)v(x)dx

= 10

u(x)v(x)dx.

A function u is called weakly differentiable with weak deriv-ative w, if every continuously differentiable function v withv(0) = 0 and v(1) = 0 satisfies1

0

w(x)v(x)dx = 1

0

u(x)v(x)dx.

Example I.2.1. Every continuously differentiable function is weaklydifferentiable and the weak derivative equals the classical derivative.Every continuous, piecewise continuously differentiable function isweakly differentiable and the weak derivative equals the classical de-rivative.The function u(x) = 1 |2x 1| is weakly differentiable with weakderivative

w(x) =

2 for 0 < x < 1

2

2 for 12

< x < 1

(cf. fig. I.2.1). Notice that the value w( 12

) is arbitrary.

I.2.4. Sobolev spaces and norms. Variational formulations andfinite element methods are based on Sobolev spaces.

The L2-Norm is defined by

v =1

0

|v(x)|2dx 1

2

.

L2(0, 1) denotes the Lebesgue space of all functions v withfinite L2-norm v.H1(0, 1) is the Sobolev space of all functions v in L2(0, 1),


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I.2. ONE-DIMENSIONAL SOBOLEV AND FINITE ELEMENT SPACES 13

dd

d

Figure I.2.1. Function u(x) = 1 |2x 1| (magenta)with its weak derivative (red)

whose weak derivative exists and is contained in L2(0, 1)too.H10 (0, 1) denotes the Sobolev space of all functions v inH1(0, 1) which satisfy v(0) = 0 and v(1) = 0.

Example I.2.2. Every bounded function is contained in L2(0, 1).The function v(x) = 1

xis not contained in L2(0, 1), since the integral

of 1x

= v(x)2 is not finite.Every continuously differentiable function is contained in H1(0, 1).

Every continuous, piecewise continuously differentiable function is con-tained in H1(0, 1).The function v(x) = 1|2x1| is contained in H10 (0, 1) (cf. fig. I.2.1).The function v(x) = 2

x is not contained in H1(0, 1), since the integral

of 1x

=

v(x))2 is not finite.

Notice that, in contrast to several dimensions, all functions inH1(0, 1) are continuous.

I.2.5. Variational formulation of the Sturm-Liouville prob-

lem. The variational formulation of the Sturm-Liouville problem is

given by:

Find u H10 (0, 1) such that for all v H10 (0, 1) there holds10

p(x)u(x)v(x) + q(x)u(x)v(x)

dx =

10

f(x)v(x)dx.

It has the following properties:

It admits a unique solution.Its solution is the unique minimum in H10 (0, 1) of the energy


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14 I. INTRODUCTION

function

1

2

10

p(x)u(x)2 + q(x)u(x)2

dx 1

0

f(x)u(x)dx.

I.2.6. Finite element spaces. The discretization of the abovevariational problem is based on finite element spaces. For their def-inition denote by T an arbitrary partition of the interval (0, 1) intonon-overlapping sub-intervals and by k 1 an arbitrary polynomialdegree.

Sk,0(T) denotes the finite element space of all continuousfunctions which are piecewise polynomials of degree k onthe intervals ofT.Sk,00 (T) is the finite element space of all functions v inSk,0(T) which satisfy v(0) = 0 and v(1) = 0.

I.2.7. Finite element discretization of the Sturm-Liouville

problem. The finite element discretization of the Sturm-Liouvilleproblem is given by:

Find a trial function uT Sk,00 (T) such that every test functionvT Sk,00 (T) satisfies1

0

p(x)uT(x)v

T(x) + q(x)uT(x)vT(x)

dx =

10

f(x)vT(x)dx.


It admits a unique solution.Its solution is the unique minimum in Sk,00 (

T) of the energy

function.After choosing a basis for Sk,00 (T) it amounts to a linearsystem of equations with k T 1 unknowns and a tridiag-onal symmetric positive definite matrix, the so-called stiff-ness matrix.Integrals are usually approximately evaluated using a quad-rature formula.In most case one chooses k = 1 (linear elements) or k = 2(quadratic elements).

One usually chooses a nodal basis for Sk,00 (T) (cf. fig. I.2.2).


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I.3. MULTI-DIMENSIONAL SOBOLEV AND FINITE ELEMENT SPACES 15

ddd

Figure I.2.2. Nodal basis functions for linear elements(left, blue) and for quadratic elements (right, endpointsof intervals blue and midpoints of intervals magenta)

I.2.8. Nodal basis functions. The nodal basis functions for lin-ear elements are those functions which take the value 1 at exactly oneendpoint of an interval and which vanish at all other endpoints of in-tervals (cf. left part of fig. I.2.2).The nodal basis functions for quadratic elements are those functions

which take the value 1 at exactly one endpoint or midpoint of an inter-val and which vanish at all other endpoints and midpoints of intervals(right part of fig. I.2.2, endpoints of intervals blue and midpoints ofintervals magenta).

I.3. Multi-dimensional Sobolev and finite element spaces

I.3.1. Domains and functions. The following notations concern-ing domains and functions will frequently be used:

open, bounded, connected set in Rd, d {2, 3}; boundary of , supposed to be Lipschitz-continuous;

D Dirichlet part of , supposed to be non-empty;N Neumann part of , may be empty;

n exterior unit normal to ;

p,q,r, . . . scalar functions with values in R;

u, v, w, . . . vector-fields with values in Rd;

S, T, . . . tensor-fields with values in Rdd;

I unit tensor;

gradient;div divergence;

div u =di=1

uixi

;

div T = di=1

Tijxi

1jd

;

= div Laplace operator;D(u) =

1

2

uixj

+ujxi 1i,jd

deformation tensor;

u v inner product;


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16 I. INTRODUCTION

S : T dyadic product (inner product of tensors).

I.3.2. Differentiation of products. The product formula for dif-ferentiation yields the following formulae for the differentiation of prod-ucts of scalar functions, vector-fields and tensor-fields:

div(pu) = p u + p div u,div(T u) = (div T) u + T : D(u).

I.3.3. Integration by parts formulae. The above product for-mulae and the Gauss theorem for integrals give rise to the followingintegration by parts formulae:

pu ndS =

p udx +

p div udx,

n T udS =

(div T) udx +

T : D(u)dx.

I.3.4. Weak derivatives. Recall that A denotes the closure of aset A Rd.Example I.3.1. For the sets

A = {x R3 : x21 + x22 + x23 < 1} open unit ballB = {x R3 : 0 < x21 + x22 + x23 < 1} punctuated open unit ballC = {x R3 : 1 < x21 + x22 + x23 < 2} open annulus

we have

A = {x R3 : x21 + x22 + x23 1} closed unit ballB = {x R3 : x21 + x22 + x23 1} closed unit ballC = {x R3 : 1 x21 + x22 + x23 2} closed annulus.

Given a continuous function : Rd R, we denote its support bysupp = {x Rd : (x) = 0}.

The set of all functions that are infinitely differentiable and have theirsupport contained in is denoted by C0 ():

C0 () = { C() : supp }.

Remark I.3.2. The condition supp is a non trivial one, sincesupp is closed and is open. Functions satisfying this condition

vanish at the boundary of together with all their derivatives.


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Given a sufficiently smooth function and a multi-index Nd,we denote its partial derivatives by

D =1+...+d

x11 . . . xdd

.

Given two functions , C0 (), the Gauss theorem for integralsyields for every multi-index Nn the identity

Ddx = (1)1+...+d

D.

This identity motivates the definition of the weak derivatives:

Given two integrable functions , L1() and a multi-index Nd, is called the -th weak derivative of ifand only if the identity

dx = (1)1+...+d

D

holds for all functions C0 (). In this case we write = D.

Remark I.3.3. For smooth functions, the notions of classical and weakderivatives coincide. However, there are functions which are not dif-ferentiable in the classical sense but which have a weak derivative (cf.Example I.3.4 below).

Example I.3.4. The function |x| is not differentiable in (1, 1), but itis differentiable in the weak sense. Its weak derivative is the piecewiseconstant function which equals 1 on (1, 0) and 1 on (0, 1).

I.3.5. Sobolev spaces and norms. We will frequently use thefollowing Sobolev spaces and norms:

Hk

() = { L2

() : D

L2

() for all Nd

with 1 + . . . + d k},

||k =

Nd1+...+d=k

D2L2() 1

2

,

k = k=0

||2 1

2

=

Nd1+...+dk

D2L2() 1

2

,

H10 () =

{

H1() : = 0 on

},


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18 I. INTRODUCTION

H1D() =

{

H1() : = 0 on D

},

H12 () = { L2() : =

for some H1()},

12, = inf{1 : H1(),

= }.

Note that all derivatives are to be understood in the weak sense.

Remark I.3.5. The space H1

2 () is called trace space of H1(), itselements are called traces of functions in H1().

Remark I.3.6. Except in one dimension, d = 1, H1 functions are in

general not continuous and do not admit point values (cf. ExampleI.3.7 below). A function, however, which is piecewise differentiable isin H1() if and only if it is globally continuous. This is crucial forfinite element functions.

Example I.3.7. The function |x| is not differentiable, but it is inH1((1, 1)). In two dimension, the function ln(ln(x21 + x22)) is anexample of an H1-function that is not continuous and which does notadmit a point value in the origin. In three dimensions, a similar exam-ple is given by ln(

x21 + x

22 + x

23).

Example I.3.8. Consider the open unit ball = {x Rd : x21 + . . . + x2d < 1}

in Rd and the functions

(x) = {x21 + . . . + x2d}2 R.

Then we have

H1()

0 if d = 2, > 1 d

2if d > 2.

I.3.6. Friedrichs and Poincare inequalities. The following in-

equalities are fundamental:

0 c||1 for all H1D(),Friedrichs inequality

0 c||1 for all H1() with

= 0

Poincare inequality.

The constants c and c depend on the domain and are propor-

tional to its diameter.


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I.3.7. Finite element partitions. The finite element discretiza-tions are based on partitions of the domain into non-overlapping

simple subdomains. The collection of these subdomains is called a par-tition and is labeled T. The members ofT, i.e. the subdomains, arecalled elements and are labeled K.

Any partition T has to satisfy the following conditions:

is the union of all elements in T. (Affine equivalence) Each K T is either a trian-

gle or a parallelogram, if d = 2, or a tetrahedronor a parallelepiped, if d = 3.

(Admissibility) Any two elements in T are eitherdisjoint or share a vertex or a complete edge or if d = 3 a complete face.

(Shape-regularity) For any element K, the ratio ofits diameter hK to the diameter K of the largestball inscribed into K is bounded independently ofK.

Remark I.3.9. In two dimensions, d = 2, shape regularity meansthat the smallest angles of all elements stay bounded away from zero.In practice one usually not only considers a single partition T, butcomplete families of partitions which are often obtained by successivelocal or global refinements. Then, the ratio hK/K must be boundeduniformly with respect to all elements and all partitions.

With every partition T we associate its shape parameter

CT = maxKT

hKK

.

Remark I.3.10. In two dimensions triangles and parallelograms may

be mixed (cf. Figure I.3.1). In three dimensions tetrahedrons andparallelepipeds can be mixed provided prismatic elements are also in-corporated. The condition of affine equivalence may be dropped. It,however, considerably simplifies the analysis since it implies constantJacobians for all element transformations.

With every partition T and its elements K we associate the follow-ing sets:

NK: the vertices of K,

EK: the edges or faces of K,


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20 I. INTRODUCTION

dd

ddd

Figure I.3.1. Mixture of triangular and quadrilateral elements

NT: the vertices of all elements in T, i.e.

NT= KTNK,

ET: the edges or faces of all elements in T, i.e.ET =

KT

EK,

NE: the vertices of an edge or face E ET,NT,: the vertices on the boundary,NT,D: the vertices on the Dirichlet boundary,NT,N: the vertices on the Neumann boundary,NT,: the vertices in the interior of ,ET,: the edges or faces contained in the boundary,ET,D: the edges or faces contained in the Dirichlet

boundary,ET,N: the edges or faces contained in the Neumann

boundary,ET,: the edges or faces having at least one endpoint

in the interior of .

For every element, face, or edge S T Ewe denote by hS itsdiameter. Note that the shape regularity of T implies that for allelements K and K and all edges E and E that share at least one

vertex the ratios hKhK , hEhE and hKhE are bounded from below and fromabove by constants which only depend on the shape parameter CT ofT.

With any element K, any edge or face E, and any vertex x weassociate the following sets (see figures I.3.2 and I.3.3)

K =EKEK=

K, K = NKNK=

K,

E =

EEKK,

E =

NENK=K,


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x = xNK K.Due to the shape-regularity ofT the diameter of any of these sets

can be bounded by a multiple of the diameter of any element or edgecontained in that set. The constant only depends on the the shapeparameter CT ofT.

d

dd

d

dd

d

ddd

dd

d

ddd

dd

dd

d

dd

d

dd

d

dd

d

dd

d

dd

d

dd

d

dd

dd

dd

dd

d

dd

d

dd

dd

dd

Figure I.3.2. Some domains K, K, E, E, and x

dd

d

dd

d

dd

d

ddd

dd

d

Figure I.3.3. Some examples of domains x

I.3.8. Finite element spaces. For any multi-index Nd weset for abbreviation

||1 = 1 + . . . + d,|| = max{i : 1 i d},

x

= x11 . . . x

dd .


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22 I. INTRODUCTION

Denote by

K = {x Rd : x1 + . . . + xd 1, xi 0, 1 i d}the reference simplex for a partition into triangles or tetrahedra andby K = [0, 1]dthe reference cube for a partition into parallelograms or parallelepipeds.

Then every element K T is the image of K under an affine mappingFK. For every integer number k set

Rk(

K) =

span{x : ||1 k} ,if K is the reference simplex,span{x : || k} ,if K is the reference cube

and setRk(K) =

p F1K : p Rk .With this notation we define finite element spaces by

Sk,1(T) = { : R : K

Rk(K) for all K T },Sk,0(T) = Sk,1(T) C(),Sk,00 (T) = Sk,0(T) H10 () = { Sk,0(T) : = 0 on }.Sk,0D (

T) = Sk,0(

T)

H1D() =

{

Sk,0(

T) : = 0 on D

}.

Note, that k may be 0 for the first space, but must be at least 1 forthe other spaces.

Example I.3.11. For the reference triangle, we have

R1(K) = span{1, x1, x2},R2(K) = span{1, x1, x2, x21, x1x2, x22}.

For the reference square on the other hand, we have

R1(K) = span{1, x1, x2, x1x2},

R2(K) = span{1, x1, x2, x1x2, x21, x21x2, x21x22, x1x22, x22}.I.3.9. Approximation properties. The finite element spaces de-

fined above satisfy the following approximation properties:

infTSk,1(T)

T0 chk+1||k+1 Hk+1(), k N,

infTSk,0(T)

| T|j chk+1j||k+1 Hk+1(),

j

{0, 1

}, k

N,


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infTS

k,00 (T) |

T|j

chk+1j

|

|k+1

Hk+1()

H10 (),

j {0, 1}, k N.

I.3.10. Nodal shape functions. Recall that NT denotes the setof all element vertices.

For any vertex x NT the associated nodal shape function is de-noted by x. It is the unique function in S

1,0(T) that equals 1 at vertexx and that vanishes at all other vertices y NT\{x}.

The support of a nodal shape function x is the set x and consists

of all elements that share the vertex x (cf. Figure I.3.3).The nodal shape functions can easily be computed element-wise

from the co-ordinates of the elements vertices.

dd

d

a0 a0a1 a1

a2 a2a3

Figure I.3.4. Enumeration of vertices of triangles andparallelograms

Example I.3.12. (1) Consider a triangle K with vertices a0, . . . , a2numbered counterclockwise (cf. Figure I.3.4). Then the restrictions toK of the nodal shape functions

a0, . . . ,

a2are given by

ai

(x) =det(x ai+1 , ai+2 ai+1)det(ai ai+1 , ai+2 ai+1) i = 0, . . . , 2,

where all indices have to be taken modulo 3.(2) Consider a parallelogram Kwith vertices a0, . . . , a3 numbered coun-terclockwise (cf. Figure I.3.4). Then the restrictions to K of the nodalshape functions

a0, . . . ,

a3are given by

ai

(x) =det(x ai+2 , ai+3 ai+2)det(ai ai+2 , ai+3 ai+2)

det(x ai+2 , ai+1 ai+2)det(ai ai+2 , ai+1 ai+2)

i = 0, . . . , 3,

where all indices have to be taken modulo 4.(3) Consider a tetrahedron K with vertices a0, . . . , a3 enumerated as in

Figure I.3.5. Then the restrictions to K of the nodal shape functions


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24 I. INTRODUCTION

a0

, . . . , a3

are given by

ai(x) =det(x

ai+1 , ai+2

ai+1 , ai+3

ai+1)

det(ai ai+1 , ai+2 ai+1 , ai+3 ai+1) i = 0, . . . , 3,where all indices have to be taken modulo 4.(4) Consider a parallelepiped K with vertices a0, . . . , a7 enumerated asin Figure I.3.5. Then the restrictions to K of the nodal shape functionsa0

, . . . , a7

are given by

ai

(x) =det(x ai+1 , ai+3 ai+1 , ai+5 ai+1)det(ai ai+1 , ai+3 ai+1 , ai+5 ai+1)

det(x ai+2 , ai+3 ai+2 , ai+6 ai+2)det(ai

ai+2 , ai+3

ai+2 , ai+6

ai+2)

det(x ai+4 , ai+5 ai+4 , ai+6 ai+4)det(ai ai+4 , ai+5 ai+4 , ai+6 ai+4)i = 0, . . . , 7,

where all indices have to be taken modulo 8.

dd

dd

dd

a0 a0a1 a1

a3

a2 a3

a7

a4

a6

a5

Figure I.3.5. Enumeration of vertices of tetrahedraand parallelepipeds (The vertex a2 of the parallelepipedis hidden.)

Remark I.3.13. For every element (triangle, parallelogram, tetrahe-dron, or parallelepiped) the sum of all nodal shape functions corre-sponding to the elements vertices is identical equal to 1 on the element.

The functions x, x NT, form a bases of S1,0(T). The basesof higher-order spaces Sk,0(T), k 2, consist of suitable products offunctions x corresponding to appropriate vertices x.

Example I.3.14. (1) Consider a again a triangle K with its vertices

numbered as in example I.3.12 (1). Then the nodal basis of S2,0(T)K

consists of the functions

ai [ai ai+1 ai+2] i = 0, . . . , 2


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4ai

ai+1

i = 0, . . . , 2,

where the functions a are as in example I.3.12 (1) and where allindices have to be taken modulo 3. An other basis of S2,0(T)

K, called

hierarchical basis, consists of the functions

ai

i = 0, . . . , 2

4ai

ai+1

i = 0, . . . , 2.

(2) Consider a again a parallelogram K with its vertices numbered as

in example I.3.12 (2). Then the nodal basis of S2,0(T)K

consists of

the functions

ai [ai ai+1 + ai+2 ai+3] i = 0, . . . , 34

ai[

ai+1

ai+2] i = 0, . . . , 3

16a0

a2

where the functions a

are as in example I.3.12 (2) and where all

indices have to be taken modulo 3. The hierarchical basis ofS2,0(T)K

consists of the functions

ai

i = 0, . . . , 3

4ai

[ai+1

ai+2

] i = 0, . . . , 3

16a0a2 .(3) Consider a again a tetrahedron K with its vertices numbered as in

example I.3.12 (3). Then the nodal basis of S2,0(T)K

consists of the

functions

ai

[ai

ai+1

ai+2

ai+3

] i = 0, . . . , 3

4ai

aj

0 i < j 3,where the functions

aare as in example I.3.12 (3) and where all

indices have to be taken modulo 4. The hierarchical basis consists ofthe functions

ai

i = 0, . . . , 3

4ai

aj

0 i < j 3.I.3.11. A quasi-interpolation operator. We will frequently use

the quasi-interpolation operator IT : L1() S1,0D (T) which is definedby

IT =

xNT,NT,N

x1

|x|x

dx.


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26 I. INTRODUCTION

Here, |x| denotes the area, if d = 2, respectively volume, if d = 3, ofthe set x.

The operator IT satisfies the following local error estimates for all H1D() and all elements K T:

ITL2(K) cA1hKH1(K), ITL2(K) cA2h

1

2

KH1(K).

Here,

K denotes the set of all elements that share at least a vertex

with K (cf. Figure I.3.6). The constants cA1 and cA2 only depend onthe shape parameter C

Tof

T.

dd

d

dd

dd

dd

dd

dd

dd

ddd

dd

dd

ddK K

dd

d

Figure I.3.6. Examples of domains

K

Remark I.3.15. The operator IT is called a quasi-interpolation op-erator since it does not interpolate a given function at the verticesx NT. In fact, point values are not defined for H1-functions. Forfunctions with more regularity which are at least in H2(), the situa-tion is different. For those functions point values do exist and the clas-sical nodal interpolation operator JT : H2() H1D() S1,0D (T) canbe defined by the relation (JT())(x) = (x) for all vertices x NT.

I.3.12. Bubble functions. For any element K T we define anelement bubble function by

K = K

xKNTx ,

K =

27 if K is a triangle,

256 if K is a tetrahedron,

16 if K is a parallelogram,

64 if K is a parallelepiped.



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0

K(x)

1 for all x

K,

K(x) = 0 for all x K,maxxK

K(x) = 1.

For every polynomial degree k there are constants cI1,k andcI2,k, which only depend on the degree k and the shape pa-rameter CT of T, such that the following inverse estimateshold for all polynomials of degree k:

cI1,kK 12

KK,

(K)

K

cI2,kh1K

K.

Recall that we denote by ET the set of all edges, ifd = 2, and of allfaces, if d = 3, of all elements in T. With each edge respectively faceE ET we associate an edge respectively face bubble function by

E = E

xENTx ,

E = 4 if E is a line segment,

27 if E is a triangle,16 if E is a parallelogram.


0 E(x) 1 for all x E,E(x) = 0 for all x E,

maxxE

E(x) = 1.

For every polynomial degree k there are constants cI3,k,cI4,k, and cI5,k, which only depend on the degree k and theshape parameter CT of T, such that the following inverseestimates hold for all polynomials of degree k:

cI3,kE 12

EE,(E)E cI4,kh

12

E E,EE cI5,kh

12

EE.


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28 I. INTRODUCTION

Here E is the union of all elements that share E (cf. Figure I.3.7).Note that E consists of two elements, if E is not contained in the

boundary , and of exactly one element, if E is a subset of .

dd

d

dd

d

dd

d

dd

d

Figure I.3.7. Examples of domains E

With each edge respectively face E ET we finally associate a unitvector nE orthogonal to E and denote by JE() the jump across E indirection nE. If E is contained in the boundary the orientation ofnE is fixed to be the one of the exterior normal. Otherwise it is notfixed.

Remark I.3.16. JE() depends on the orientation ofnE but quantitiesof the form JE(nE ) are independent of this orientation.


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CHAPTER II

A posteriori error estimates

In this chapter we will describe various possibilities for a posteriorierror estimation. In order to keep the presentation as simple as possi-ble we will consider in sections II.1 and II.2 a simple model problem:the two-dimensional Poisson equation (cf. equation (II.1.1) (p. 30))

discretized by continuous linear or bilinear finite elements (cf. equation(II.1.3) (p. 30)). We will review several a posteriori error estimatorsand show that in a certain sense they are all equivalent and yieldlower and upper bounds on the error of the finite element discretization.The estimators can roughly be classified as follows:

Residual estimates: Estimate the error of the computed nu-merical solution by a suitable norm of its residual with respectto the strong form of the differential equation (section II.1.9(p. 38)).

Solution of auxiliary local problems: On small patches of

elements, solve auxiliary discrete problems similar to, but sim-pler than the original problem and use appropriate norms ofthe local solutions for error estimation (section II.2.1 (p. 40)).

Hierarchical basis error estimates: Evaluate the residual ofthe computed finite element solution with respect to anotherfinite element space corresponding to higher order elements orto a refined grid (section II.2.2 (p. 46)).

Averaging methods: Use some local extrapolate or average ofthe gradient of the computed numerical solution for error es-timation (section II.2.3 (p. 51)).

Equilibrated residuals: Evaluate approximately a dual varia-

tional problem posed on a function space with a weaker topol-ogy (section II.2.4 (p. 53)).

H(div)-lifting: Sweeping through the elements sharing a givenvertex construct a vector field such that its divergence equalsthe residual (section II.2.5 (p. 57)).

In section II.2.6 (p. 60), we shortly address the question of asymptoticexactness, i.e., whether the ratio of the estimated and the exact errorremains bounded or even approaches 1 when the mesh-size convergesto 0. In section II.2.7 (p. 62) we finally show that an adaptive methodbased on a suitable error estimator and a suitable mesh-refinement

strategy converges to the true solution of the differential equation.29


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30 II. A POSTERIORI ERROR ESTIMATES

II.1. A residual error estimator for the model problem

II.1.1. The model problem. As a model problem we considerthe Poisson equation with mixed Dirichlet-Neumann boundary condi-tions

u = f in u = 0 on D(II.1.1)

u

n= g on N

in a connected, bounded, polygonal domain

R2 with boundary

consisting of two disjoint parts D and N. We assume that the Dirich-let boundary D is closed relative to and has a positive length andthat f and g are square integrable functions on and N, respectively.The Neumann boundary N may be empty.

II.1.2. Variational formulation. The standard weak formula-tion of problem (II.1.1) is:

Find u H1D() such that

(II.1.2) u v = f v + N gvfor all v H1D().

It is well-known that problem (II.1.2) admits a unique solution.

II.1.3. Finite element discretization. As in section I.3.7 (p.19) we choose an affine equivalent, admissible and shape-regular parti-tion Tof . We then consider the following finite element discretizationof problem (II.1.2):

Find uT S1,0D (T) such that(II.1.3)

uT vT =

f vT+

N

gvT

for all vT S1,0D (T).

Again it is well-known that problem (II.1.3) admits a unique solution.

II.1.4. Equivalence of error and residual. In what follows wealways denote by u

H1D() and uT

S1,0D (

T) the exact solutions of

problems (II.1.2) and (II.1.3), respectively. They satisfy the identity


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II.1. A RESIDUAL ERROR ESTIMATOR 31

(u uT) v = f v + N gv uT vfor all v H1D(). The right-hand side of this equation implicitlydefines the residual of uT as an element of the dual space of H1D().

The Friedrichs and Cauchy-Schwarz inequalities imply for all v H1D()

11 + c2

v1 supwH1

D()w1=1

v w v1.

This corresponds to the fact that the bilinear form

H1D() v, w

v w

defines an isomorphism of H1D() onto its dual space. The constantsmultiplying the first and last term in this inequality are related to thenorm of this isomorphism and of its inverse.

The definition of the residual and the above inequality imply theestimate

supwH1

D()w1=1

f w + N

gw

uT w u uT1

1 + c2 supwH1

D()w1=1

f w +

N

gw

uT w

.

Since the sup-term in this inequality is equivalent to the norm of theresidual in the dual space of H1D(), we have proved:

The norm in H1D() of the error is, up to multiplicativeconstants, bounded from above and from below by the normof the residual in the dual space of H1D().

Most a posteriori error estimators try to estimate this dual norm ofthe residual by quantities that can more easily be computed from f, g,and uT.

II.1.5. Galerkin orthogonality. Since S1,0D (

T)

H1D(), the

error is orthogonal to S1,0D (T):


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(u uT) wT = 0for all wT S1,0D (T). Using the definition of the residual, this can bewritten as

f wT+

N

gwT

uT wT = 0

for all wT S1,0D (T). This identity reflects the fact that the discretiza-tion (II.1.3) is consistent and that no additional errors are introducedby numerical integration or by inexact solution of the discrete problem.It is often referred to as Galerkin orthogonality.

II.1.6. L2-representation of the residual. Integration by partselement-wise yields for all w H1D()

f w +

N

gw

uT w

=

f w +

N

gw

KTK

uT w

=

f w +

N

gw +KT

K

uTw K

nK uTw=KT

K

(f + uT)w +

EET,N

E

(g nE uT)w

EET,

E

JE(nE uT)w.

Here, nK denotes the unit exterior normal to the element K. Note thatuT

vanishes on all triangles.

For abbreviation, we define element and edge residuals by

RK(uT) = f + uT

and

RE(uT) =

JE(nE uT) if E ET,,g nE uT if E ET,N,0 if E ET,D .

Then we obtain the following L2-representation of the residual


33/144


f w + N gw uT w=KT

K

RK(uT)w +EET

E

RE(uT)w.

Together with the Galerkin orthogonality this implies

f w +

N

gw

uT w

= KTK

RK(uT)(w wT)

+EET

E

RE(uT)(w wT)

for all w H1D() and all wT S1,0D (T).

II.1.7. Upper error bound. We fix an arbitrary function wH1D() and choose wT = ITw with the quasi-interpolation operator of

section I.3.11 (p. 25). The Cauchy-Schwarz inequality for integralsand the properties of IT then yield

f w +

N

gw

uT w

=KT

K

RK(uT)(w ITw) +EET

E

RE(uT)(w ITw)

KTRK(uT)Kw ITwK + EETRE(uT)Ew ITwEKT

RK(uT)KcA1hKw1,K +EET

RE(uT)EcA2h1

2

Ew1,E .

Invoking the Cauchy-Schwarz inequality for sums this gives

f w +

N

gw

uT w

max

{cA1, cA2

}KTh2K

RK(uT

)

2K


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+ EET hERE(uT)2E12

KT

w21,K +EET

w21,E 1

2

.

In a last step we observe that the shape-regularity of T impliesKT

w21,K +EET

w21,E 1

2

cw1

with a constant c which only depends on the shape parameter CT of

T and which takes into account that every element is counted severaltimes on the left-hand side of this inequality.Combining these estimates with the equivalence of error and resid-

ual, we obtain the following upper bound on the error

u uT1 cKT

h2KRK(uT)2K

+

EEThERE(uT)2E

12

with

c =

1 + c2 max{cA1, cA2}c.The right-hand side of this estimate can be used as an a posteri-

ori error estimator since it only involves the known data f and g, thesolution uT of the discrete problem, and the geometrical data of thepartition. The above inequality implies that the a posteriori error es-timator is reliable in the sense that an inequality of the form errorestimator tolerance implies that the true error is also less than thetolerance up to the multiplicative constant c. We want to show thatthe error estimator is also efficient in the sense that an inequality ofthe form error estimator tolerance implies that the true error isalso greater than the tolerance possibly up to another multiplicativeconstant.

For general functions f and g the exact evaluation of the integralsoccurring on the right-hand side of the above estimate may be prohibi-tively expensive or even impossible. The integrals then must be approx-imated by suitable quadrature formulae. Alternatively the functions fand g may be approximated by simpler functions, e.g., piecewise poly-nomial ones, and the resulting integrals be evaluated exactly. Often,

both approaches are equivalent.


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II.1.8. Lower error bound. In order to prove the announcedefficiency, we denote for every element K by fK the mean value of f

on K

fK =1

|K|K

f dx

and for every edge E on the Neumann boundary by gE the mean valueof g on E

gE =1

|E

| EgdS.

We fix an arbitrary element K and insert the function

wK = (fK + uT)K

in the L2-representation of the residual. Taking into account thatsupp wK K we obtain

K

RK(uT)wK =K

(u uT) wK.

We add

K(fK f)wK on both sides of this equation and obtain

K

(fK + uT)2K = K

(fK + uT)wK

=

K

(u uT) wKK

(f fK)wK.

The results of section I.3.12 (p. 26) imply for the left hand-side of thisequation

K

(fK + uT)2K c2I1fK + uT2K

and for the two terms on its right-hand sideK

(u uT) wK (u uT)KwKK (u uT)KcI2h1K fK + uTK

K

(f fK)wK f fKKwKK f fKKfK + uTK.

This proves that

hKfK + uTK c2I1 cI2(u uT)K+ c

2I1 hKf fKK.

(II.1.4)


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Next, we consider an arbitrary interior edge E ET, and insert thefunction

wE = RE(uT)Ein the L2-representation of the residual. This gives

E

JE(nE uT)2E =E

RE(uT)wE

=

E

(u uT) wE

KTEEK

K

RK(uT)wE

= E

(u uT) wE

KTEEK

K

(fK + uT)wE

KTEEK

K

(f fK)wE

The results of section I.3.12 (p. 26) imply for the left-hand side of thisequation

E

JE(nE uT)2E c2I3JE(nE uT)2E

and for the three terms on its right-hand sideE

(u uT) wE (u uT)1,EwE1,E (u uT)1,E

cI4h1

2

E JE(nE uT)E

KTEEK

K

(fK + uT)wE KTEEK

fK + uTKwEK

KTEEK

fK + uTK

cI5h1

2

EJE(nE uT)E

KTEEK K(f fK)wE

KTEEKf fKKwEK


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Invoking once again the results of section I.3.12 (p. 26) and using thesame arguments as above this implies that

h1

2

EgE nE uTE c2I3 cI5

cI4 + c2I1 cI2 (u uT)K

+ c2I3 cI5

1 + c2I1

hKf fKK+ c2I3 h

1

2

Eg gEE.

(II.1.6)

Estimates (II.1.4), (II.1.5), and (II.1.6) prove the announced efficiencyof the a posteriori error estimate:

h2KfK + uT2K+

1

2

EEKET,

hEJE(nE uT)2E

+

EEKET,NhEgE nE uT2E

12

cu uT21,K+

KTEKEK=h2Kf fK21,K

+

EEKET,NhEg gE2E

12

.

The constant c only depends on the shape parameter CT.

II.1.9. Residual a posteriori error estimate. The results ofthe preceding sections can be summarized as follows:

Denote by u

H1D() and u

T S1,0D (

T) the unique so-

lutions of problems (II.1.2) (p. 30) and (II.1.3) (p. 30),respectively. For every element K T define the residuala posteriori error estimator R,K by

R,K =

h2KfK + uT2K+

1

2

EEKET,

hEJE(nE uT)2E

+

EEKET,N

hEgE nE uT2E 1

2

,


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where fK and gE are the mean values of f and g on K

and E, respectively. There are two constants c and c,which only depend on the shape parameter CT, such thatthe estimates

u uT1 cKT

2R,K

+KT

h2Kf fK2K

+

EET,NhEg gE2E

12

and

R,K cu uT21,K+KTEKEK=

h2Kf fK21,K

+

EEKET,NhEg gE2E

12

hold for all K

T.

Remark II.1.1. The factor 12

multiplying the second term in R,Ktakes into account that each interior edge is counted twice when addingall 2R,K. Note that uT = 0 on all triangles.

Remark II.1.2. The first term in R,K is related to the residual ofuTwith respect to the strong form of the differential equation. The secondand third term in R,K are related to that boundary operator which iscanonically associated with the strong and weak form of the differentialequation. These boundary terms are crucial when considering low orderfinite element discretizations as done here. Consider e.g. problem(II.1.1) (p. 30) in the unit square (0, 1)2 with Dirichlet boundaryconditions on the left and bottom part and exact solution u(x) = x1x2.When using a triangulation consisting of right angled isosceles trianglesand evaluating the line integrals by the trapezoidal rule, the solutionof problem (II.1.3) (p. 30) satisfies uT(x) = u(x) for all x NT butuT = u. The second and third term in R,K reflect the fact that uT /H2() and that uT does not exactly satisfy the Neumann boundarycondition.

Remark II.1.3. The correction terms

hKf fKK and h1

2Eg gEE


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in the above a posteriori error estimate are in general higher order per-turbations of the other terms. In special situations, however, they can

be dominant. To see this, assume that T contains at least one triangle,choose a triangle K0 T and a non-zero function 0 C0 (

K0), and

consider problem (II.1.1) (p. 30) with f = 0 and D = . SinceK0

f = K0

0 = 0

and f = 0 outside K0, we have

fK = 0

for all K T. Since

f vT =

K0

0vT = K0

0vT = 0

for all vT S1,0D (T), the exact solution of problem (II.1.3) (p. 30) isuT = 0.

Hence, we have

R,K = 0

for all K T, butu uT1 = 0.

This effect is not restricted to the particular approximation off consid-

ered here. Since 0 C0 ( K0) is completely arbitrary, we will alwaysencounter similar difficulties as long as we do not evaluate fK ex-actly which in general is impossible. Obviously, this problem is curedwhen further refining the mesh.

II.2. A catalogue of error estimators for the model problem

II.2.1. Solution of auxiliary local discrete problems. Theresults of section II.1 show that we must reliably estimate the normof the residual as an element of the dual space of H1D(). This couldbe achieved by lifting the residual to a suitable subspace of H1

D

()by solving auxiliary problems similar to, but simpler than the originaldiscrete problem (II.1.3) (p. 30). Practical considerations and theresults of the section II.1 suggest that the auxiliary problems shouldsatisfy the following conditions:

In order to get an information on the local behaviour of theerror, they should involve only small subdomains of .

In order to yield an accurate information on the error, theyshould be based on finite element spaces which are more ac-curate than the original one.

In order to keep the computational work at a minimum, they

should involve as few degrees of freedom as possible.


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I I.2. A CATALOGUE OF ERROR ESTIMATORS 41

To each edge and eventually to each element there should cor-respond at least one degree of freedom in at least one of the

auxiliary problems. The solution of all auxiliary problems should not cost more

than the assembly of the stiffness matrix of problem (II.1.3)(p. 30).

There are many possible ways to satisfy these conditions. Here, wepresent three of them. To this end we denote by P1 = span{1, x1, x2}the space of linear polynomials in two variables.

II.2.1.1. Dirichlet problems associated with vertices. First, we de-cide to impose Dirichlet boundary conditions on the auxiliary problems.The fourth condition then implies that the corresponding subdomains

must consist of more than one element. A reasonable choice is to con-sider all nodes x NT, NT,N and the corresponding domains x(see figures I.3.2 (p. 21) and I.3.3 (p. 21)). The above conditions thenlead to the following definition:Set for all x NT, NT,N

Vx = span{K, E, E : K T, x NK,E ET,, x NE,E ET,N, E x,,,

P1

}and

D,x = vxx

where vx Vx is the unique solution of

xvx w = KTxNK K

fKw + EET,NEx EgEw

x

uT w

for all w Vx.In order to get a different interpretation of the above problem, set

ux = uT+ vx.

Then

D,x = (ux uT)x


42/144


and ux uT+ Vx is the unique solution of

x

ux w = KTxNK

K

fKw + EET,NEx

E

gEw

for all w Vx. This is a discrete analogue of the following Dirichletproblem

= f in x = uT on x\N

n= g on x N.

Hence, we can interpret the error estimator D,x in two ways: We solve a local analogue of the residual equation using a

higher order finite element approximation and use a suitablenorm of the solution as error estimator.

We solve a local discrete analogue of the original problem usinga higher order finite element space and compare the solutionof this problem to the one of problem (II.1.3) (p. 30).

Thus, in a certain sense, D,x is based on an extrapolation technique.It can be shown that it yields upper and lower bounds on the erroru

u

Tand that it is comparable to the estimator R,T.

Denote by u H1D() and uT S1,0D () the unique so-lutions of problems (II.1.2) (p. 30) and (II.1.3) (p. 30).There are constants cN,1, . . . , cN,4, which only depend onthe shape parameter CT, such that the estimates

D,x cN,1

KTxNK

2R,K

12

,

R,K cN,2 xNK\NT,D 2D,x

1

2

,

D,x cN,3u uT21,x+KTxNK

h2Kf fK2K

+

EET,NEx

hEg gE2E 1

2

,

u

uT1

cN,4 xNT,NT,N

2

D,x


43/144


+KTh

2

Kf fK2

K

+

EET,N

hEg gE2E 1

2

hold for all x NT, NT,N and all K T. Here, fK,gE, and R,K are as in sections II.1.8 (p. 35) and II.1.9 (p.38).

II.2.1.2. Dirichlet problems associated with elements. We now con-sider an estimator which is a slight variation of the preceding one.

Instead of all x NT, NT,N and the corresponding domains x weconsider all K T and the corresponding sets K (see figure I.3.2 (p.21)). The considerations from the beginning of this section then leadto the following definition:Set for all K T

VK = span{K, E, E : K T, EK EK = ,E EK ET,,E ET,N, E K,,, P1}

and

D,K = vKKwhere vK VK is the unique solution of

K vK w = KTEKEK=KfKw + EET,N

EKE gEw

K

uT w

for all w VK.As before we can interpret uT+ vK as an approximate solution of

the following Dirichlet problem

= f in K = uT on K\N


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n= g on K N.

It can be shown that D,K also yields upper and lower bounds on theerror u uT and that it is comparable to D,x and R,K.

Denote by u H1D() and uT S1,0D (T) the unique solu-tions of problem (II.1.2) (p. 30) and (II.1.3) (p. 30). Thereare constants cT,1, . . . , cT,4, which only depend on the shapeparameter CT, such that the estimates

D,K cT,1

KTEKEK=2R,K

1

2

,

R,K cT,2

KTEKEK=

2D,K 1

2

,

D,K cT,3u uT21,K+KTEKEK=

h2Kf fK2K

+ EET,NEK

hE

g gE

2

E12

,

u uT1 cT,4KT

2D,K

+KT

h2Kf fK2K

+

EET,N

hEg gE2E 1

2

hold for all K T. Here, fK, gE, R,K are as in sectionsII.1.8 (p. 35) and II.1.9 (p. 38).

II.2.1.3. Neumann problems. For the third estimator we decide toimpose Neumann boundary conditions on the auxiliary problems. Nowit is possible to choose the elements in T as the corresponding subdo-main. This leads to the definition:Set for alle K T

VK = span{K, E : E EK\ET,D , , P1}


45/144


and

N,K = vKK

where vK is the unique solution ofK

vK w =K

(fK + uT)w

12

EEKET,

E

JE(nE uT)w

+ EEKET,N

E(gE nE uT)wfor all w VK.

Note, that the factor 12

multiplying the residuals on interior edgestakes into account that interior edges are counted twice when summingthe contributions of all elements.

The above problem can be interpreted as a discrete analogue of thefollowing Neumann problem

= RK(u

T) in K

n=

1

2RE(uT) on K

n= RE(uT) on K N

= 0 on K D.Again it can be shown that N,K also yields upper and lower bounds

on the error and that it is comparable to R,K.

Denote by u H1D() and uT S1,0D (T) the unique solu-tions of problem (II.1.2) (p. 30) and (II.1.3) (p. 30). Thereare constants cT,5, . . . , cT,8, which only depend on the shapeparameter CT, such that the estimates

N,K cT,5R,K,

R,K cT,6

KTEKEK=

2N,K 1

2

,

N,K cT,7u uT21,K+ KTEKEK=

h2K

f

fK

2K


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+ EEKET,N

hEg gE2

E12

,

u uT1 cT,8KT

2N,K

+KT

h2Kf fK2K

+

EET,N

hEg gE2E 1

2

hold for all K T

. Here, fK, gE, R,K are as in sectionsII.1.8 (p. 35) and II.1.9 (p. 38).

Remark II.2.1. When T exclusively consists of triangles uT van-ishes element-wise and the normal derivatives nE uT are edge-wiseconstant. In this case the functions , , and can be dropped in thedefinitions ofVx, VK, and VK. This considerably reduces the dimensionof the spaces Vx, VK, and VK and thus of the discrete auxiliary prob-lems. Figures I.3.2 (p. 21) and I.3.3 (p. 21) show typical examples ofdomains x and K. From this it is obvious that in general the aboveauxiliary discrete problems have at least the dimensions 12, 7, and 4,respectively. In any case the computation of D,x, D,K, and N,K ismore expensive than the one of R,K. This is sometimes payed off byan improved accuracy of the error estimate.

II.2.2. Hierarchical error estimates. The key-idea of the hier-archical approach is to solve problem (II.1.2) (p. 30) approximatelyusing a more accurate finite element space and to compare this solutionwith the solution of problem (II.1.3) (p. 30). In order to reduce thecomputational cost of the new problem, the new finite element space isdecomposed into the original one and a nearly orthogonal higher ordercomplement. Then only the contribution corresponding to the com-plement is computed. To further reduce the computational cost, theoriginal bilinear form is replaced by an equivalent one which leads to adiagonal stiffness matrix.

To describe this idea in detail, we consider a finite element spaceYT which satisfies S1.0D (T) YT H1D() and which either consists ofhigher order elements or corresponds to a refinement of T. We thendenote by wT YT the unique solution of

(II.2.1)

wT vT =

f vT+

N

gvT

for all vT YT.


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To compare the solutions wT of problem (II.2.1) and uT of problem(II.1.3) (p. 30) we subtract uT vT on both sides of equation(II.2.1) and take the Galerkin orthogonality into account. We thusobtain

(wT uT) vT =

f vT+

N

gvT

uT vT

=

(u uT) vTfor all vT YT, where u H1D() is the unique solution of problem(II.1.2) (p. 30). Since S1.0D (T) YT, we may insert vT = wT uTas a test-function in this equation. The Cauchy-Schwarz inequality for

integrals then implies(wT uT) (u uT).To prove the converse estimate, we assume that the space YT satis-

fies a saturation assumption, i.e., there is a constant with 0 < 1such that

(II.2.2) (u wT) (u uT).From the saturation assumption (II.2.2) and the triangle inequality weimmediately conclude that

(u

u

T)

(u

w

T)

+

(w

Tu

T)

(u uT) + (wT uT)and therefore

(u uT) 11 (wT uT).

Thus, we have proven the two-sided error bound

(wT uT) (u uT) 1

1

(wT uT).

Hence, we may use (wT uT) as an a posteriori error estimator.This device, however, is not efficient since the computation of wT

is at least as costly as the one of uT. In order to obtain a more efficienterror estimation, we use a hierarchical splitting

YT = S1,0D (T) ZT

and assume that the spaces S1,0D (T) and ZT are nearly orthogonal andsatisfy a strengthened Cauchy-Schwarz inequality, i.e., there is a con-stant with 0 < 1 such that(II.2.3)

| vT zT| vTzT


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holds for all vT S1,0D (T), zT ZT.Now, we write w

TuT

in the form vT

+ zT

with vT

S1,0D (

T) and

zT ZT. From the strengthened Cauchy-Schwarz inequality we thendeduce that

(1 ){vT2 + zT2} (wT uT)2 (1 + ){vT2 + zT2}

and in particular

zT 11

(wT uT).(II.2.4)

Denote by zT ZT the unique solution of(II.2.5)

zT T =

f T+

N

gT

uT Tfor all T ZT.

From the definitions (II.1.2) (p. 30), (II.1.3) (p. 30), (II.2.1), and(II.2.5) of u, uT, wT, and zT we infer that

zT T =

(u uT) T(II.2.6)

= (wT uT) Tfor all T ZT and

(wT uT) vT = 0(II.2.7)

for all vT S1,0D (T). We insert T = zT in equation (II.2.6). TheCauchy-Schwarz inequality for integrals then yields

zT (u uT).On the other hand, we conclude from inequality (II.2.4) and equations

(II.2.6) and (II.2.7) with T = zT that

(wT uT)2 =

(wT uT) (wT uT)

=

(wT uT) (vT+ zT)

=

(wT uT) zT

=

zT zT

zTzT


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11

zT(wT uT)

and hence

(u uT) 11 (wT uT)

1(1 )1 zT.

Thus, we have established the two-sided error bound

zT (u uT) 1

(1 )1 zT.

Therefore, zT can be used as an error estimator.At first sight, its computation seems to be cheaper than the one of

wT since the dimension of ZT is smaller than that of YT. The com-putation of zT, however, still requires the solution of a global systemand is therefore as expensive as the calculation of uT and wT. But, inmost applications the functions in ZT vanish at the vertices ofNT sinceZT is the hierarchical complement of S

1,0D (T) in YT. This in particu-

lar implies that the stiffness matrix corresponding to ZT

is spectrally

equivalent to a suitably scaled lumped mass matrix. Therefore, zTcan be replaced by a quantity zT which can be computed by solving adiagonal linear system of equations.

More precisely, we assume that there is a bilinear form b on ZTZTwhich has a diagonal stiffness matrix and which defines an equivalentnorm to on ZT, i.e.,(II.2.8) T2 b(T, T) T2

holds for all T ZT with constants 0 < .The conditions on b imply that there is a unique function z

T ZTwhich satisfies

(II.2.9) b(zT, T) =

f T+

N

gT

uT T

for all T ZT.The Galerkin orthogonality and equation (II.2.5) imply

b(zT, T) =

(u uT) T

= zT T


50/144


for all T ZT. Inserting T = zT and T = zT in this identity andusing estimate (II.2.8) we infer that

b(zT, zT) =

(u uT) zT (u uT)zT (u uT) 1

b(zT, zT)

12

and

zT2 = b(zT, zT)

b(z

T, z

T)12 b(zT, zT)

12

b(zT, zT) 12zT.This proves the two-sided error bound

b(zT, zT)

12 (u uT)

(1 )1 b(zT, zT)

12 .

We may summarize the results of this section as follows:

Denote by u H1D() and uT S1,0D (T) the unique so-lutions of problems (II.1.2) (p. 30) and (II.1.3) (p. 30),respectively. Assume that the space YT = S

1,0D (T)ZT sat-

isfies the saturation assumption (II.2.2) and the strength-ened Cauchy-Schwarz inequality (II.2.3) and admits a bilin-ear form b on ZTZT which has a diagonal stiffness matrixand which satisfies estimate (II.2.8). Denote by zT ZTthe unique solution of problem (II.2.9) and define the hier-archical a posteriori error estimator H by

H = b(zT, zT)

12 .

Then the a posteriori error estimates

(u uT) (1 )1 H

and

H 1(u uT)

are valid.

Remark II.2.2. When considering families of partitions obtained bysuccessive refinement, the constants and in the saturation as-

sumption and the strengthened Cauchy-Schwarz inequality should be


51/144


uniformly less than 1. Similarly, the quotient

should be uniformlybounded.

Remark II.2.3. The bilinear form b can often be constructed as fol-lows. The hierarchical complement ZT can be chosen such that itselements vanish at the element vertices NT. Standard scaling argu-ments then imply that on ZT the H1-semi-norm is equivalent toa scaled L2-norm. Similarly, one can then prove that the mass-matrixcorresponding to this norm is spectrally equivalent to a lumped mass-matrix. The lumping process in turn corresponds to a suitable numer-ical quadrature. The bilinear form b then is given by the inner-productcorresponding to the weighted L2-norm evaluated with the quadraturerule.

Remark II.2.4. The strengthened Cauchy-Schwarz inequality, e.g.,holds if YT consists of continuous piecewise quadratic or biquadraticfunctions. Often it can be established by transforming to the referenceelement and solving a small eigenvalue-problem there.

Remark II.2.5. The saturation assumption (II.2.2) is used to estab-lish the reliability of the error estimator H. One can prove that thereliability of H in turn implies the saturation assumption (II.2.2). Ifthe space YT contains the functions wK and wE of section II.1.8 (p.

35) one may repeat the proofs of estimates (II.1.4) (p. 35), (II.1.5) (p.37), and (II.1.6) (p. 38) and obtains that up to perturbation terms

of the form hKf fKK and h1

2

Eg gEE the quantity zTK isbounded from below by R,K for every element K. Together with theresults of section II.1.9 (p. 38) and inequality (II.2.9) this proves upto the perturbation terms the reliability of H without resorting tothe saturation assumption. In fact, this result may be used to showthat the saturation assumption holds if the right-hand sides f and gof problem (II.1.1) (p. 30) are piecewise constant on T and ET,N,respectively.

II.2.3. Averaging techniques. To avoid unnecessary technicaldifficulties and to simplify the presentation, we consider in this sec-tion problem (II.1.1) (p. 30) with pure Dirichlet boundary conditions,i.e. N = , and assume that the partition T exclusively consists oftriangles.

The error estimator of this chapter is based on the following ideas.Denote by u and uT the unique solutions of problems (II.1.2) (p. 30)and (II.1.3) (p. 30). Suppose that we dispose of an easily computableapproximation GuT ofuT such that

(II.2.10) u GuT u uT


52/144


holds with a constant 0 < 1. We then have1

1 + GuT uT u uT 1

1 GuT uT

and may therefore choose GuT uT as an error estimator. SinceuT is a piecewise constant vector-field we may hope that its L2-projection onto the continuous, piecewise linear vector-fields satisfiesinequality (II.2.10). The computation of this projection, however, isas expensive as the solution of problem (II.1.3) (p. 30). We thereforereplace the L2-scalar product by an approximation which leads to amore tractable auxiliary problem.

In order to make these ideas more precise, we denote by WT thespace of all piecewise linear vector-fields and set VT = WT C(,R2).Note that XT WT. We define a mesh-dependent scalar product(, )T on WT by

(v, w)T =KT

|K|3

xNK

v|K(x) w|K(x)

.

Here, |K| denotes the area of K and|K(x) = limyx

yK

(y)

for all WT, K T, x NK.Since the quadrature formula

K

|K|3

xNK

(x)

is exact for all linear functions, we have

(II.2.11) (v, w)T =

v wif both arguments are elements of W

Tand at least one of them is

piecewise constant. Moreover, one easily checks that1

4v2 (v, v)T v2

for all v WT and

(v, w)T =1

3

xNT

|x|v(x) w(x)(II.2.12)

for all v, w VT.Denote by GuT VT the (, )T-projection ofuT onto VT, i.e.,

(GuT, vT)T = (uT, vT)T


53/144


for all vT VT. Equations (II.2.11) and (II.2.12) imply that

GuT(x) = KTxNK

|K||x|uT|K

for all x NT. Thus, GuT may be computed by a local averaging ofuT.

We finally set

Z,K = GuT uTKand

Z = KT

2Z,K 12 .One can prove that Z yields upper and lower bounds for the error

and that it is comparable to the residual error estimator R,K of sectionII.1.9 (p. 38).

II.2.4. Equilibrated residuals. The error estimator of this sec-tion is due to Ladeveze and Leguillon and is based on a dual variationalprinciple.

We define the energy norm || corresponding to the variationalproblem (II.1.2) (p. 30) by

|v|2 =

v v

and a quadratic functional J on H1D() by

J(v) =1

2

v v

f v

N

gv +

uT v,

where uT S1,0D (T) is the unique solution of problem (II.1.3) (p. 30).The Galerkin orthogonality implies that

J(v) =1

2

v v

(u uT) v

for all v H1D(). Hence, J attains its unique minimum at u uT.Inserting v = u uT in the definition of J therefore yields

12|u uT|2 = J(u uT) J(v)

for all v H1D(). Thus, the energy norm of the error can be computedby solving the variational problem

minimize J(v) in H1D().


54/144


Unfortunately, this is an infinite dimensional minimization problem.In order to obtain a more tractable problem we want to replace H1D()

by the broken space

HT = {v L2() : v|K H1(K) for all K T, v = 0 on D}.Obviously, we have

H1D() = {v HT : JE(v) = 0 for all E ET,},where JE(v) denotes the jump of v across E in direction nE.

One can prove that a continuous linear functional on HT vanisheson H1D() if and only if there is a vector-field L2()d with div L2() and n = 0 on N such that

(v) = KTK

nK v

for all v VT. Hence, the polar of H1D() in VT can be identified withthe space

M = { L2()d : div L2(), n = 0 on N}.We want to extend the residual R(uT) of section II.1.4 (p. 30) to

a continuous linear functional on the larger space HT. To this end weassociate with each E ET a smooth vector-field E. The choice ofEis arbitrary subject to the constraint that E = g

|E for all E

ET,N.

The particular choice of the fluxes E for the interelement boundarieswill later on determine the error estimation method; for E D thevalue of E is completely irrelevant. Once we have chosen the E wecan associate with each element K T a vector-field K defined onK such that

(II.2.13)KT

K

K v =EET

E

E JE(v)

for all v HT. Here, we use the convention that JE(v) = v if E .We then define the extension R(uT) of R(uT) to HT by R(uT), v =

KT

K

f v K

uT v +K

Kv

EET,

E

EJE(v)

for all v HT.Due to equation (II.2.13) and the definition of E on the Neumann

boundary N, we have

R(uT), v = f v + N gv uT v


55/144


for all v H1D(). Moreover, there is a M such that

(v) = EET,

E

EJE(v)

for all v HT since EET,

E

EJE(v) = 0

holds for all v H1D().Now, we define a Lagrangian functional L on HT M by

L(v, ) =1

2 KTKv v RT(uT), v (v).M is the space of Langrange multipliers for the constraint

JE(v) = 0 for all E ET,.This implies that

12|u uT|2 = inf

vH1D

()J(v)

= inf wHT

supM

L(w, )

= supM infwHT L(w, ).

Hence, we get for all M

12|u uT|2 inf

wHTL(w, )

= inf wHT

KT

12

K

w w K

f w

+ KuT w KKw+ (w) (w)

.

The particular choice = therefore yields

|u uT|2 2 infwHT

KT

1

2

K

w w K

f w

+ KuT w K Kw.


56/144


In order to write this in a more compact form, we denote by HK therestriction of H1D() to a single element K

Tand set

JK(w) =1

2

K

w w K

f w +K

uT w K

Kw

for all w HK and all K T. We then haveHT =

KT

HK

and therefore

(II.2.14) |u uT|2 2KT

infwKHK

JK(wK).

Estimate (II.2.14) reduces the computation of the energy norm ofthe error to a family of minimization problems on the elements in T.However, for each K T, the corresponding variational problem stillis infinite dimensional. In order to overcome this difficulty we firstrewrite JK. Using integration by parts we see that

JK(w) =1

2

K

w w K

(f + uT)w

K

(K nK uT)

=1

2 Kw w KRK(uT)wK

RK(uT) w,

where

RK(uT) = f + uT

and

RK(uT) = K nK uT.

For every element K T we defineYK = { L2(K)d : div L2(K),

div = RK(uT), nK = RK(uT)}.(II.2.15)

The complementary energy principle then tells us that

infwHK

JK(w) = supYK

12

K

.

Together with inequality (II.2.14) this implies that

|u

uT|

2

2KT supKYK

1

2 KK K


57/144


=

KTinf

KYK KK K.

This is the announced dual variational principle. It proves:

Denote by u H1D() and uT S1,0D () the unique so-lutions of problems (II.1.2) (p. 30) and (II.1.3) (p. 30),respectively. For every element K T define the space YKby equation (II.2.15). Then the a posteriori error estimate

(u uT)2 KT

K

K K

holds for all K YK and all K T.

Remark II.2.6. The concrete realization of the equilibrated residualmethod depends on the choice of the K and on the definition of the K.Ladeveze and Leguillon choose K to be the average of nK uT fromthe neighbouring elements plus a suitable piecewise linear function onK. The functions K are often chosen as the solution of a maximiza-tion problem on a finite dimensional subspace of YK corresponding tohigher order finite elements. With a proper choice of K and K onemay thus recover the error estimator N,K of section II.2.1.3 (p. 44).

II.2.5. H(div)-lifting. The basic idea is to construct a piece-wiselinear vector field T such that

(II.2.16)

div T = f on every K TJE(nE T) = JE(nE uT) on every E E

n T = g n uT on every E EN.Then the vector field = T + uT is contained in H(div;) andsatisfies

(II.2.17) div = f in

n = g on N.since uT vanishes element-wise.

To simplify the presentation we assume for the rest of this sectionthat

T exclusively consists of triangles, f is piece-wise constant, g is piece-wise constant.

Parallelograms could be treated by changing the definition (II.2.18) ofthe vector fields K,E. General functions f and g introduce additional

data errors.


58/144


For every triangle K and every edge E thereof we denote by aK,Ethe vertex of K which is not contained in E and set

(II.2.18) K,E(x) =1(E)

22(K)(x aK,E).

The vector fields K,E are the shape functions of the lowest orderRaviart-Thomas space and have the following properties

(II.2.19)

div K,E =1(E)

2(K)on K,

nK K,E = 0 on K \ E,nK K,E = 1 on E,

K,E

K

chK,

where nK denotes the unit exterior normal ofKand where the constantc only depends on the shape parameter ofT.

dd

ddd

K1

K2K3

K4

K5

K6

K7

Figure II.2.1. Enumeration of elements in z

Now, we consider an arbitrary interior vertex z N. We enumer-ate the triangles in z from 1 to n and the edges emanating from zfrom 0 to n such that (cf. figure II.2.1)

E0 = En, Ei1 and Ei are edges of Ki for every i.

We define

0 = 0

and recursively for i = 1, . . . , n

i = 2(Ki)31(Ei)

f +1(Ei1)21(Ei)

JEi1(nEi1 uT) +1(Ei1)

1(Ei)i1.

By induction we obtain

1(En)n = ni=1

2(Ki)

3f +

n1j=0

1(Ej)

2JEj (nEj uT).

Since

Ki z =2(Ki)

3


59/144


for every i {1, . . . , n} and since

Ej

z = 1(Ej)2

for every j {0, . . . , n 1}, we conclude using the assumption thatf and g are piece-wise constant that

ni=1

2(Ki)

3f +

n1j=0

1(Ej)

2JEj(nEj uT) =

rz

jz

= 0.

Hence we have n = 0. Therefore we can define a vector field z by

setting for every i {1, . . . , n}(II.2.20) z|Ki = iKi,Ei

JEi1(nEi1 uT) + i1

Ki,Ei1.

Equations (II.2.19) and the definition of the i imply that

(II.2.21) div z = 1

3f on Ki

JEi(z nEi) = 1

2JE(nEi uT) on Ei

holds for every i {1, . . . , n}.For a vertex on the boundary , the construction of z must be

modified as follows:

For every edge on the Neumann boundary N we must replaceJE(nE uT) by g n uT.

Ifz is a vertex on the Dirichlet boundary, there is at least oneedge emanating from z which is contained in D. We mustchoose the enumeration of the edges such that En is one ofthese edges.

With these modifications, equations (II.2.20) and (II.2.21) carry overalthough in general n = 0 for vertices on the boundary .

In a final step, we extend the vector fields z by zero outside z and

set

(II.2.22) T =zN

z.

Since every triangle has three vertices and every edge has two vertices,we conclude from equations (II.2.21) that T has the desired properties(II.2.16).

The last inequality in (II.2.19), the definition of the i, and theobservation that uT vanishes element-wise imply that

z

z

cKz h

2K

f + u

T2K


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+

EzhEJE(nE uT)2E

+

EzNhEg nE uT2E

12

holds for every vertex z N with a constant which only depends onthe shape parameter ofT.

Combining these results, we arrive at the following a posteriori errorestimates:

(u uT) T

T

c

(u

uT

)

II.2.6. Asymptotic exactness. The quality of an a posteriorierror estimator is often measured by its efficiency index, i.e., the ratioof the estimated error and of the true error. An error estimator is calledefficient if its efficiency index together with its inverse remain boundedfor all mesh-sizes. It is called asymptotically exact if its efficiency indextends to one when the mesh-size converges to zero.

In the generic case we have

KTh

2

Kf fK2

K12

= o(h)

and EET,N

hEg gE2E 1

2

= o(h),

where

h = maxKT

hK

denotes the maximal mesh-size. On the other hand, the solutions ofproblems (II.1.2) (p. 30) and (II.1.3) (p. 30) satisfy

u uT1 chalways but in trivial cases. Hence, the results of sections II.1.9 (p.38), II.2.1.1 (p. 41), II.2.1.2 (p. 43), II.2.1.3 (p. 44), II.2.3 (p. 51),and II.2.3 (p. 51) imply that the corresponding error estimators areefficient. Their efficiency indices can in principle be estimated explicitlysince the constants in the above sections only depend on the constantsin the quasi-interpolation error estimate of section I.3.11 (p. 25) andthe inverse inequalities of section I.3.12 (p. 26) for which sharp boundscan be derived.

Using super-convergence results one can also prove that on special

meshes the error estimators of sections II.1.9 (p. 38), II.2.1.1 (p. 41),


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II.2.1.2 (p. 43), II.2.1.3 (p. 44), II.2.3 (p. 51), and II.2.3 (p. 51) areasymptotically exact.

The following example shows that asymptotic exactness may nothold on general meshes even if they are strongly structured.

dd

d

dd

dd

dd

dd

dd

dd

d

dd

dd

dd

dd

dd

dd

dd

dd

dd

dd

ddd

dd

dd

dd

dd

d

Figure II.2.2. Triangulation of example II.2.7 corre-sponding to n = 4

Example II.2.7. Consider problem (II.1.1) (p. 30) on the unit square

= (0, 1)2

with

N = (0, 1) {0} (0, 1) {1},g = 0,

and

f = 1.

The exact solution is

u(x, y) =1

2x(1 x).

The triangulation

Tis obtained as follows (cf. figure II.2.2): is

divided into n2 squares with sides of length h = 1n , n N; eachsquare is cut into four triangles by drawing the two diagonals. Thistriangulation is often called a criss-cross grid. Since the solution uof problem (II.1.1) (p. 30) is quadratic and the Neumann boundaryconditions are homogeneous, one easily checks that the solution uT ofproblem (II.1.3) (p. 30) is given by

uT(x) =

u(x) if x is a vertex of a square,

u(x) h224

if x is a midpoint of a square.

Using this expression for uT one can explicitly calculate the error and

the error estimator. After some computations one obtains for any


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square Q, which is disjoint from N,

KTKQ

2N,K 12eQ =176 1.68.Hence, the error estimator cannot be asymptotically exact.

II.2.7. Convergence. Assume that we dispose of an error esti-mator K which yields global upper and local lower bounds for theerror of the solution of problem (II.1.2) (p. 30) and its finite elementdiscretization (II.1.3) (p. 30) and that we apply the general adaptivealgorithm I.1.1 (p. 7) with one of the refinement strategies of algo-rithms III.1.1 (p. 97) and III.1.2 (p. 98). Then one can prove that theerror decreases linearly. More precisely: If u denotes the solution ofproblem (II.1.2) and if ui denotes the solution of the discrete problem(II.1.3) corresponding to the i-the partition Ti, then there is a constant0 < < 1, which only depends on the constants c and c in the errorbounds, such that

(u ui) i(u u0).

II.3. Elliptic problems

II.3.1. Scalar linear elliptic equations. In this section we con-sider scalar linear elliptic partial differential equations in their generalform

div(Au) + a u + u = f in u = 0 on D

n Au = g on N

where the diffusion A(x) is for every x a symmetric positive definitematrix. We assume that the data satisfy the following conditions: The diffusion A is continuously differentiable and uniformly

elliptic and uniformly isotropic, i.e.,

= infx

minzRd\{0}

ztA(x)z

ztz> 0

and

= 1 supx

maxzRd\{0}

ztA(x)z

ztz

is of moderate size.


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II.3. ELLIPTIC PROBLEMS 63

The convection a is a continuously differentiable vector fieldand scaled such that

supx

|a(x)| 1.

The reaction is a continuous non-nega

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