Fast Solvers for Higher Order Problems · Elliptic Problems in Non Smooth Domains (1985) Dauge...

Fast Solvers forHigher Order Problems

Susanne C. Brenner

Department of Mathematicsand

Center for Computation & Technology

Louisiana State University

Susanne C. Brenner IMA Workshop Fast Solution Techniques

Fast Solvers for Higher Order Problems

Outline

I Model Problems

I Discretizations

I Multigrid Methods

I Domain Decomposition Methods

I Adaptive Methods

I Concluding Remarks

Collaborators: Thirupathi Gudi, Shiyuan Gu, Michael Neilan,Li-yeng Sung and Kening Wang



Model Problems


Fast Solvers for Higher Order Problems Model Problems

Biharmonic Equation

∆2u = f in Ω

with different boundary conditions.

Ω = bounded polygonal domain in R2

∆ =∂2

∂x21

+∂2

∂x22



Biharmonic Equation

∆2u = f in Ω



∆ =∂2

∂x21

+∂2

∂x22

Boundary Conditions of Clamped Plates

u = 0 on ∂Ω∂u∂n

= 0 on ∂Ω



Biharmonic Equation

∆2u = f in Ω



∆ =∂2

∂x21

+∂2

∂x22

Boundary Conditions of Simply Supported Plates

u = 0 on ∂Ω∆u = 0 on ∂Ω



Biharmonic Equation

∆2u = f in Ω



∆ =∂2

∂x21

+∂2

∂x22

Boundary Conditions of the Cahn-Hilliard Type

∂u∂n

= 0 on ∂Ω

∂(∆u)∂n

= 0 on ∂Ω



Elliptic Regularity

Since ∂Ω is not smooth, the solutions of the model problemshave limited regularity even if

f ∈ C∞(Ω)

The Shift Theorem fails in general for f ∈ L2(Ω), i.e.,

f ∈ L2(Ω) does not imply u ∈ H4(Ω)



Elliptic Regularity


f ∈ C∞(Ω)



There exists α ≤ 2, depending on the interior angles of Ω andthe boundary conditions, such that the solution u of the modelproblem belongs to H2+α(Ω) when f ∈ L2(Ω) and we have

‖u‖H2+α(Ω) ≤ CΩ,α‖f‖L2(Ω)



Elliptic Regularity


f ∈ C∞(Ω)



There exists α ≤ 2, depending on the interior angles of Ω andthe boundary conditions, such that the solution u of the modelproblem belongs to H2+α(Ω) when f ∈ L2(Ω) and we have

‖u‖H2+α(Ω) ≤ CΩ,α‖f‖L2(Ω)

α = index of elliptic regularity



Elliptic Regularity

Boundary Conditions of Clamped Plates

u =∂u∂n

= 0 on ∂Ω

The index of elliptic regularity α satisfies

12< α ≤ 2

α > 1 if Ω is convex.

α < 1 if Ω is non-convex.



Elliptic Regularity


u = ∆u = 0 on ∂Ω


0 < α < 2

For an equilateral triangle, α can be any number < 2.

For a rectangle, α can be any number < 1.

For an L-shaped domain, α can be any number < 13 .

α can be close to 0 if there is an interior angle of Ω close to π.



Elliptic Regularity


∂u∂n

=∂(∆u)∂n

= 0 on ∂Ω

Under the compatibility condition∫Ω

f dx = 0

the boundary value problem is solvable and there is a uniquesolution that satisfies ∫

Ωu dx = 0



Elliptic Regularity


∂u∂n

=∂(∆u)∂n

= 0 on ∂Ω


0 < α < 2

and has the same behavior as the index of elliptic regularity forsimply supported plates.



Elliptic Regularity

ReferencesBlum and RannacherOn the boundary value problem of the biharmonic operator ondomains with angular corners

Math. Methods Appl. Sci. (1980)

GrisvardElliptic Problems in Non Smooth Domains (1985)

DaugeElliptic Boundary Value Problems on Corner Domains (1988)

Kozlov, Maz’ya and RossmannSpectral Problems Associated with Corner Singularities of Solu-tions to Elliptic Problems (2001)



Difficulties



DifficultiesI Conforming methods require C1 elements (Argyris,

Hseih-Clough-Tocher, Bogner-Fox-Schmit, ...) which arequite complicated (more so in 3D).





I Classical nonconforming methods (Morley, de Veubeke,...) do not come in natural hierarchies and hence are notefficient for capturing smoother solutions.






I Moreover it requires a lot of ingenuity to design noncon-forming methods that work and very little is known aboutnonconforming elements in 3D.






I Moreover it requires a lot of ingenuity to design noncon-forming methods that work and very little is known aboutnonconforming elements in 3D.

I Mixed finite element methods, which split the biharmonicequation into two Laplace equations, work well for theclamped plate boundary conditions. But they do not workfor the boundary conditions of simply supported platesand those of the Cahn-Hilliard type when the domain isnon-convex.



Difficulties

I Actually the situation is even worse. The mixed formu-lation can miss the leading singularity and the solutionsobtained by such mixed methods will converge to a wrongsolution (Sapondzhyan paradox).



Difficulties


ReferenceNazarov and Plamenevsky

Elliptic Problems in Domains with Piecewise SmoothBoundaries (1994)



Difficulties


I It is also not easy to find finite element pairs that satisfythe L-B-B inf-sup condition for stability.



Difficulties



I Even if you have the correct mixed formulation and a sta-ble finite element pair, at the end you still need to solve asaddle point problem, while the original problem is SPD.



Difficulties



I Even if you have the correct mixed formulation and a sta-ble finite element pair, at the end you still need to solve asaddle point problem, while the original problem is SPD.

I Furthermore the adaptation of these methods (conform-ing, nonconforming and mixed) to domains with curvedboundaries is not straight-forward.



Difficulties

I Another major difficulty is that the discrete problems aremore ill-conditioned than the discrete problems for sec-ond order problems. The condition number grows at theorder of O(h−4) instead of O(h−2) for second order prob-lems.



Difficulties

I Another major difficulty is that the discrete problems aremore ill-conditioned than the discrete problems for sec-ond order problems. The condition number grows at theorder of O(h−4) instead of O(h−2) for second order prob-lems.

I We need discretizations that can overcome the draw-backs of the standard methods and at the same time ad-mit fast solvers that can overcome the ill-conditioning ofthe discrete problems.



Discretizations


Fast Solvers for Higher Order Problems Discretizations

Problem of Clamped Plates

Ω = bounded polygonal domain f ∈ L2(Ω)

∆2u = f in Ω

u =∂u∂n

= 0 on ∂Ω



Problem of Clamped Plates

Ω = bounded polygonal domain f ∈ L2(Ω)

∆2u = f in Ω

u =∂u∂n

= 0 on ∂Ω

Weak Formulation Find u ∈ H20(Ω) such that

a(u, v) =∫

Ωfv dx ∀ v ∈ H2

0(Ω)

a(w, v) =∫

ΩD2w : D2v dx D2w : D2v =

2∑i,j=1

wxixjvxixj



Finite Element Spaces

Th = a simplicial triangulation of Ω

Vh (⊂ H10(Ω)) = Pk (k ≥ 2) Lagrange finite element space

k=3k=2






k=3k=2

We can also use Qk Lagrange finite element spaces.

k=3k=2






k=3k=2

These are standard C0 finite element spaces for second orderproblems.



Discrete Problem

Find uh ∈ Vh such that

Ah(uh, v) =∫

Ωfv dx ∀ v ∈ Vh

Ah(w, v) =∑T∈Th

∫T

D2w : D2v dx +∑e∈Eh

∫e

∂2w∂n2

[[∂v∂n

]]ds

+∑e∈Eh

∫e

∂2v∂n2

[[∂w∂n

]]ds

+ σ∑e∈Eh

1|e|

∫e

[[∂w∂n

]][[∂v∂n

]]ds

Eh = set of edges · = average [[·]] = jump

|e| = length of e σ = penalty parameter



Discrete Problem

This is an interior penalty method obtained through integrationby parts, symmetrization and stabilization.



Discrete Problem

This is an interior penalty method obtained through integrationby parts, symmetrization and stabilization.

Ah(w, v) =∑T∈Th

∫T

D2w : D2v dx︸︷︷︸piecewise version of continuous variational form (ibp)

+∑e∈Eh

∫e

∂2w∂n2

[[∂v∂n

]]ds︸︷︷︸

consistency (ibp)

+∑e∈Eh

∫e

∂2v∂n2

[[∂w∂n

]]ds︸︷︷︸

symmetrization

+∑e∈Eh

σ

|e|

∫e

[[∂w∂n

]] [[∂v∂n

]]ds︸︷︷︸

stabilization



Discrete Problem

Since the finite element functions are globally continuous, thisis a C0 interior penalty method. It is a discontinuous Galerkinmethod for fourth order problems, where the discontinuity isin the normal derivative across element boundaries. The dis-crete problem is a SPD problem when the penalty parameteris sufficiently large. Therefore it preserves the SPD propertyof the continuous problem.



Discrete Problem

Since the finite element functions are globally continuous, thisis a C0 interior penalty method. It is a discontinuous Galerkinmethod for fourth order problems, where the discontinuity isin the normal derivative across element boundaries. The dis-crete problem is a SPD problem when the penalty parameteris sufficiently large. Therefore it preserves the SPD propertyof the continuous problem.

Reference

G. Engel, K. Garikipati, T.J.R. Hughes, M.G. Larson, L. Mazzei, and R.L.Taylor

Continuous/discontinuous finite element approximations of fourth orderelliptic problems in structural and continuum mechanics with applicationsto thin beams and plates, and strain gradient elasticity

Comput. Methods Appl. Mech. Engrg. (2002)



Error Analysis: First Approach




I Show that C0 interior penalty methods are consistent, i.e.,the solution u of the continuous problem satisfies

Ah(u, v) =∫


Since the solution u does not belong to H4(Ω) in general,this requires a careful analysis to justify the integration byparts that involves u.




I Show that C0 interior penalty methods are consistent.

I Use the norm ||| · |||h defined by

|||v|||2h =∑T∈Th

|v|2H2(T) +∑e∈Eh

σ

|e|‖[[∂v/∂n]]‖2

L2(e)

+∑e∈Eh

|e|σ‖∂2v/∂n2‖2

L2(e)






|||v|||2h =∑T∈Th

|v|2H2(T) +∑e∈Eh

σ

|e|‖[[∂v/∂n]]‖2

L2(e)

+∑e∈Eh

|e|σ‖∂2v/∂n2‖2

L2(e)

This norm is not well-defined on H2(Ω), where the contin-uous problem is posed.






|||v|||2h =∑T∈Th

|v|2H2(T) +∑e∈Eh

σ

|e|‖[[∂v/∂n]]‖2

L2(e)

+∑e∈Eh

|e|σ‖∂2v/∂n2‖2

L2(e)

I Show that Ah(·, ·) is bounded, i.e.,

|Ah(w, v)| ≤ C|||w|||h|||v|||h ∀ v,w ∈ 〈u〉+ Vh

This requires that ‖∂2u/∂n2‖L2(e) <∞.(true for the clamped plates since u ∈ H2+α(Ω) for some α > 1/2)






|||v|||2h =∑T∈Th

|v|2H2(T) +∑e∈Eh

σ

|e|‖[[∂v/∂n]]‖2

L2(e)

+∑e∈Eh

|e|σ‖∂2v/∂n2‖2

L2(e)

I Show that Ah(·, ·) is bounded, i.e.,

|Ah(w, v)| ≤ C|||w|||h|||v|||h ∀ v,w ∈ 〈u〉+ Vh

I Show that Ah(·, ·) is coercive on Vh for σ 1, i.e.,

|Ah(v, v)| ≥ β|||v|||2h ∀ v ∈ Vh (β a positive constant)




I We can then show that C0 interior penalty methods arequasi-optimal in the norm ||| · |||h, i.e.

|||u− uh|||h ≤ C infv∈Vh|||u− v|||h




I We can then show that C0 interior penalty methods arequasi-optimal in the norm ||| · |||h, i.e.

|||u− uh|||h ≤ C infv∈Vh|||u− v|||h

I It follows that

‖u− uh‖h ≤ C|||u−Πhu|||h ≤ Chα‖f‖L2(Ω)

where Πh : C0(Ω) −→ Vh is the nodal interpolation opera-tor for the Lagrange finite element and α ∈ (1/2, 2] is theindex of elliptic regularity for clamped plates:

‖u‖H2+α(Ω) ≤ C‖f‖L2(Ω)




I In the case where the solution u is smooth (u ∈ H`(Ω))and higher order finite elements (k ≥ `− 1) are used, wehave a better error estimate

‖u− uh‖h ≤ Ch`−2|u|H`(Ω)

i.e., higher order C0 interior penalty methods can capturesmooth solutions efficiently.




I In the case where the solution u is smooth (u ∈ H`(Ω))and higher order finite elements (k ≥ `− 1) are used, wehave a better error estimate

‖u− uh‖h ≤ Ch`−2|u|H`(Ω)

Details can be found in the following paper

ReferenceB. and Sung

C0 interior penalty methods for fourth order ellipticboundary value problems on polygonal domains

J. Sci. Comput. (2005)




I This standard approach requires that u ∈ H2+α(Ω) forsome α > (1/2) so that

‖∂2u/∂n2‖L2(e) <∞





‖∂2u/∂n2‖L2(e) <∞

I This elliptic regularity holds for the biharmonic equa-tion with the boundary conditions of clamped plates.But it does not hold on general polygonal domains forthe boundary conditions of simply supported plates andthose of the Cahn-Hilliard type.

u = ∆u = 0 (simply supported plates) 0 < α < 2∂u∂n

=∂(∆u)∂n

= 0 (Cahn-Hilliard) 0 < α < 2





‖∂2u/∂n2‖L2(e) <∞

I This elliptic regularity holds for the biharmonic equa-tion with the boundary conditions of clamped plates.But it does not hold on general polygonal domains forthe boundary conditions of simply supported plates andthose of the Cahn-Hilliard type.

u = ∆u = 0 (simply supported plates) 0 < α < 2∂u∂n

=∂(∆u)∂n

= 0 (Cahn-Hilliard) 0 < α < 2

I The standard approach is therefore problematic for suchproblems.



Error Analysis: Second Approach




I In this approach we use the norm ‖ · ‖h given by

‖v‖2h =

∑T∈Th

|v|2H2(T) +∑e∈Eh

σ

|e|‖[[∂v/∂n]]‖2

L2(e)

(well-defined on H2(Ω))





‖v‖2h =

∑T∈Th

|v|2H2(T) +∑e∈Eh

σ

|e|‖[[∂v/∂n]]‖2

L2(e)


I We do not use the Galerkin orthogonality of the C0 interiorpenalty methods. Hence there is no need to use ellipticregularity to justify integration by parts that involves u.





‖v‖2h =

∑T∈Th

|v|2H2(T) +∑e∈Eh

σ

|e|‖[[∂v/∂n]]‖2

L2(e)


I We do not use the Galerkin orthogonality of the C0 interiorpenalty methods. Hence there is no need to use ellipticregularity to justify integration by parts that involves u.

I In this approach the whole error analysis can be carriedout using only the facts that u belongs to H2

0(Ω) and thatu satisfies the weak form of the boundary value problem.




I It can be shown that

‖u− uh‖h ≤ C(

infv∈Vh‖u− v‖h + Osc(f )

)where

Osc(f ) =( ∑

T∈Th

h4T inf

q∈Pk−2(T)‖f − q‖2

L2(T)

)1/2

is of higher order. (k = degree of the polynomials in Vh)

(quasi-optimal up to a higher order term)




I It can be shown that

‖u− uh‖h ≤ C(

infv∈Vh‖u− v‖h + Osc(f )

)where

Osc(f ) =( ∑

T∈Th

h4T inf

q∈Pk−2(T)‖f − q‖2

L2(T)

)1/2

is of higher order. (k = degree of the polynomials in Vh)

I Now we can use elliptic regularity to obtain

‖u− uh‖h ≤ C(‖u−Πhu‖h + Osc(f )

)≤ Chα‖f‖L2(Ω)

where α is the index of elliptic regularity.




I The proof uses the Berger-Scott-Strang Lemma for non-conforming methods from a priori error analysis and bub-ble function techniques from a posteriori error analysis. Itcan therefore be called a medius error analysis




I The proof uses the Berger-Scott-Strang Lemma for non-conforming methods from a priori error analysis and bub-ble function techniques from a posteriori error analysis. Itcan therefore be called a medius error analysis

I Details can be found in the following paper.

ReferenceGudiA new error analysis for discontinuous finite elementmethods for linear elliptic problems

Math. Comp. (2010)




I This new approach can be applied to any discontinuousfinite element method, including classical nonconformingmethods and discontinuous Galerkin methods, for sec-ond and fourth order problems.





I It puts the analysis of such methods on an equal footingwith the analysis of conforming methods. First we derivean abstract quasi-optimal error estimate using only theweak formulation of the boundary value problem. Thenwe use elliptic regularity to obtain a concrete error esti-mate.





I It puts the analysis of such methods on an equal footingwith the analysis of conforming methods. First we derivean abstract quasi-optimal error estimate using only theweak formulation of the boundary value problem. Thenwe use elliptic regularity to obtain a concrete error esti-mate.

I It is also an efficient way to perform both a priori anda posteriori error analyses, since the local efficiency ofresidual based error estimators is already a part of themedius analysis.




I This second approach makes it possible to analyze C0

interior penalty methods for the biharmonic equation withthe boundary conditions of simply supported plates andthose of the Cahn-Hilliard type.






Ah(uh, v) =∫


Ah(w, v) =∑T∈Th

∫T

D2w : D2v dx +∑e∈E i

h

∫e

∂2w∂n2

[[∂v∂n

]]ds

+∑e∈E i

h

∫e

∂2v∂n2

[[∂w∂n

]]ds

+ σ∑e∈E i

h

1|e|

∫e

[[∂w∂n

]][[∂v∂n

]]ds

E ih = set of interior edges (same finite element space Vh ⊂ H1

0(Ω))






Ah(uh, v) =∫


Ah(w, v) =∑T∈Th

∫T

D2w : D2v dx +∑e∈Eh

∫e

∂2w∂n2

[[∂v∂n

]]ds

+∑e∈Eh

∫e

∂2v∂n2

[[∂w∂n

]]ds

+ σ∑e∈Eh

1|e|

∫e

[[∂w∂n

]][[∂v∂n

]]ds

Vh ⊂ H1(Ω) with zero mean (same bilinear form)




I Details can be found in the following papers.

ReferenceB. and NeilanA C0 interior penalty method for a fourth order ellipticsingular perturbation problem

preprint 2010B., Gu, Gudi and Sung

A C0 interior penalty method for a biharmonic problemwith essential and natural boundary conditions of Cahn-Hilliard type

preprint 2010




I The medius analysis has also been applied to sixth orderelliptic boundary value problems.

ReferenceGudi and NeilanAn interior penalty method for a sixth order elliptic equa-tion

IMA J. Numer. Anal. (to appear)



Enriching Operator Eh

We can connect the C0 finite element space Vh to H20(Ω) by

an operator Eh that maps Vh to a C1 finite element space Vh(⊂H2

0(Ω)). Such an operator can be constructed by averaging.

Vh = Pk f.e. space −→ Vh = Argyris f.e. space

Vh = Qk f.e. space −→ Vh = BFS f.e. space




We can connect the C0 finite element space Vh to H20(Ω) by

an operator Eh that maps Vh to a C1 finite element space Vh(⊂H2

0(Ω)). Such an operator can be constructed by averaging.

Vh = Pk f.e. space −→ Vh = Argyris f.e. space

Vh = Qk f.e. space −→ Vh = BFS f.e. space

Estimates for Eh∑T∈Th

(h−4

T ‖v− Ehv‖2L2(T) + h−2

T |v− Ehv|2H1(T) + |Ehv|2H2(T)

)≤ C

∑e∈Eh

1|e|‖[[v]]‖2

L2(e) ≤ C‖v‖2h ∀ v ∈ Vh




Estimates for Eh Πh∑T∈Th

(h−2(2+α)

T ‖ζ − EhΠhζ‖2L2(T) + h−2(1+α)

T |ζ − EhΠhζ|2H1(T)

+ h−2αT |ζ − EhΠhζ|2H2(T)

)≤ C|ζ|2H2+α(Ω) ∀ ζ ∈ H2+α(Ω) ∩ H2

0(Ω)

(Eh Πh behaves like a quasi-local interpolation operator.)




Estimates for Eh Πh∑T∈Th

(h−2(2+α)

T ‖ζ − EhΠhζ‖2L2(T) + h−2(1+α)

T |ζ − EhΠhζ|2H1(T)

+ h−2αT |ζ − EhΠhζ|2H2(T)

)≤ C|ζ|2H2+α(Ω) ∀ ζ ∈ H2+α(Ω) ∩ H2

0(Ω)

Let uh ∈ Vh be the solution obtained by a C0 interior penaltymethod. Then Ehuh ∈ Vh ∈ H2

0(Ω) is a post-processed C1

finite element solution for the biharmonic equation. We canshow, using the estimates for Eh and Eh Πh, that the post-processed solution Ehuh also satisfies quasi-optimal error es-timates. Therefore C0 interior penalty methods are also rele-vant for computing H2(Ω) solutions.




The enriching operator also plays an important role in the de-sign and analysis of fast solvers for the discrete problems re-sulting from C0 interior penalty methods.



Features of C0 Interior Penalty Methods




I The lowest order C0 interior penalty methods are as sim-ple as the classical nonconforming methods.





I Smooth solutions can be captured efficiently by higherorder C0 interior penalty methods, which are simpler thanhigher order C1 finite element methods.






I Unlike mixed methods, C0 interior penalty methods canbe extended in a straight-forward way (integration byparts, symmetrization and stabilization) to more compli-cated fourth order problems. They also preserve the SPDproperty of the continuous problem.






I Unlike mixed methods, C0 interior penalty methods canbe extended in a straight-forward way (integration byparts, symmetrization and stabilization) to more compli-cated fourth order problems. They also preserve the SPDproperty of the continuous problem.

I In situations where the solution has low regularity, errorestimates for C0 interior penalty methods can be estab-lished by the medius analysis.




I There are enriching operators connecting the C0 finite el-ement spaces to the Sobolev spaces where the contin-uous problem is posed. These operators satisfy manyuseful estimates.





I Since C0 interior penalty methods are based on finite ele-ment spaces for second order problems, the isoparamet-ric approach works perfectly for domains with a curvedboundary.

ReferenceB., Neilan and Sung

Isoparametric C0 interior penalty methods for platebending problems (in preparation)





I Since C0 interior penalty methods are based on finite ele-ment spaces for second order problems, the isoparamet-ric approach works perfectly for domains with a curvedboundary.

I Since the finite element spaces are standard spaces forsecond order problems, it is easy to implement Poissonsolves as preconditioners, which reduce the conditionnumber from O(h−4) to O(h−2).

(more on this later)



Multigrid Methods


Fast Solvers for Higher Order Problems Multigrid Methods

Set-Up

Tk = triangulations of Ω obtained by regular subdivision(k = 0, 1, 2, . . .)

hk = mesh size of Tk (hk = 2 hk+1)

Vk = Pj/Qj (j ≥ 2) finite element space associated with Tk

V0 ⊂ V1 ⊂ · · ·Vk ⊂ Vk+1 ⊂ · · ·



Set-Up

k-th Level Discrete Problem (clamped plates)

Find uk ∈ Vk such that

Ak(uk, v) =∫

Ωfv dx ∀ v ∈ Vk (f ∈ L2(Ω))

Ak(w, v) =∑T∈Tk

∫T

D2w : D2v dx +∑e∈Ek

∫e

∂2w∂n2

[[∂v∂n

]]ds

+∑e∈Ek

∫e

∂2v∂n2

[[∂w∂n

]]ds

+∑e∈Ek

σ

|e|

∫e

[[∂w∂n

]] [[∂v∂n

]]ds



Set-Up

Discrete Differential Operator Ak : Vk −→ V ′k

〈Akv,w〉 = Ak(v,w) ∀ v,w ∈ Vk

〈·, ·〉 = canonical bilinear form on V ′k × Vk

The matrix representing Ak with respect to the natural nodalbasis of the finite element space Vk and the dual basis of V ′kis the stiffness matrix, whose condition number is O(h−4

k ).



Set-Up

Discrete Differential Operator Ak : Vk −→ V ′k

〈Akv,w〉 = Ak(v,w) ∀ v,w ∈ Vk

〈·, ·〉 = canonical bilinear form on V ′k × Vk

k-th Level Discrete Problem

Find uk ∈ Vk such that

Akuk = φk

〈φk, v〉 =∫

Ωfv dx ∀ v ∈ Vk



Features of Multigrid

Multigrid algorithms are iterative methods for the system

(∗) Akz = ψ

where z ∈ Vk and ψ ∈ V ′k.





(∗) Akz = ψ


I optimal complexity

I performance independent of the number of grid levels

computational cost proportional to the number of unknowns





(∗) Akz = ψ


I optimal complexityI performance independent of the number of grid levels

The system (∗) is very ill-conditioned for large k (small h)and the convergence of classical iterative methods be-comes very slow. But multigrid can overcome the ill-conditioning of (∗).





(∗) Akz = ψ


I optimal complexityI performance independent of the number of grid levels

I two ingredients in the design of multigrid methods

− intergrid transfer operators

− a smoothing scheme

to damp out the highly oscillatory part of the error so that the remaining part can be captured accurately

on a coarser grid



Intergrid Transfer Operators

V0 ⊂ V1 ⊂ · · ·Vk−1 ⊂ Vk ⊂ · · ·

Coarse-to-Fine Operator Ikk−1 : Vk−1 −→ Vk

Ikk−1 = natural injection



Intergrid Transfer Operators

V0 ⊂ V1 ⊂ · · ·Vk−1 ⊂ Vk ⊂ · · ·

Coarse-to-Fine Operator Ikk−1 : Vk−1 −→ Vk

Ikk−1 = natural injection

Fine-to-Coarse Operator Ik−1k : V ′k −→ V ′k−1

〈Ik−1k ψ, v〉 = 〈ψ, Ik

k−1v〉

for all ψ ∈ V ′k and v ∈ Vk−1



Smoother for Akz = ψ (z ∈ Vk, ψ ∈ V ′k)

znew = zold + ωkB−1k (ψ − Akzold)

B−1k : V ′k −→ Vk is an SPD operator which is an approximate

inverse of the discrete Laplace operator Lk : Vk −→ V ′k definedby

〈Lkv1, v2〉 =∫

Ω∇v1 · ∇v2 dx ∀ v ∈ Vk

ωk = damping factor

(a preconditioned Richardson relaxation scheme)





B−1k : V ′k −→ Vk is an SPD operator which is an approximate

inverse of the discrete Laplace operator Lk : Vk −→ V ′k definedby

〈Lkv1, v2〉 =∫

Ω∇v1 · ∇v2 dx ∀ v ∈ Vk

ωk = damping factor

We can take B−1k to be a multigrid Poisson solve, which can be

easily implemented because the finite element spaces in theC0 interior penalty methods are standard spaces for secondorder problems.

(A MG algorithm for second order problems is embedded in the MG algorithm for fourth order problems.)





B−1k : V ′k −→ Vk is an approximate inverse of the discrete

Laplace operator.

Properties of the Preconditioner

Spectral Radius of B−1k Ak

ρ(B−1k Ak) ≈ h−2

k

Damping Factorωk = Ch2

k (ρ(ωkB−1k Ak) ≤ 1)





B−1k : V ′k −→ Vk is an approximate inverse of the discrete

Laplace operator.

Properties of the Preconditioner

Spectral Radius of B−1k Ak

ρ(B−1k Ak) ≈ h−2

k

Damping Factorωk = Ch2

k (ρ(ωkB−1k Ak) ≤ 1)

Condition Number of B−1k Ak

κ(B−1k Ak) ≈ h−2

k κ(Ak) ≈ h−4k



Multigrid Algorithms for Akz = ψ (z ∈ Vk, ψ ∈ V ′k)




For k = 0 we use a direct solve to solve the equation exactly.

For k > 0 the algorithm is defined recursively in three steps.




pre-smoothing

apply m preconditioned relaxation steps with initial guess z0 toobtain the approximate solution z∗




pre-smoothing


coarse grid correction

apply the (k−1)-st level iteration scheme p times to the coarsegrid residual equation

Ak−1e = Ik−1k (ψ − Akz∗)

with initial guess 0 to obtain the correction ek−1 ∈ Vk−1




pre-smoothing






post-smoothing

apply m preconditioned relaxation steps with initial guessz∗ + Ik

k−1ek−1

to obtain the final output




pre-smoothing






post-smoothing

apply m preconditioned relaxation steps with initial guessz∗ + Ik

k−1ek−1

to obtain the final output




If in the coarse grid correction step we use one iteration of the(k−1)-st level scheme (p = 1), we have the V-cycle algorithm.

If in the coarse grid correction step we use two iterations of the(k−1)-st level scheme (p = 2), we have the W-cycle algorithm.

If in the coarse grid correction step we use one iteration of the(k−1)-st level scheme followed by one iteration of the (k−1)-stlevel V cycle scheme, we have the F-cycle algorithm.




If in the coarse grid correction step we use one iteration of the(k−1)-st level scheme (p = 1), we have the V-cycle algorithm.

If in the coarse grid correction step we use two iterations of the(k−1)-st level scheme (p = 2), we have the W-cycle algorithm.

If in the coarse grid correction step we use one iteration of the(k−1)-st level scheme followed by one iteration of the (k−1)-stlevel V cycle scheme, we have the F-cycle algorithm.

For nonconforming methods the F-cycle algorithm is more ro-bust than the V-cycle algorithm and it performs almost as wellas the W-cycle algorithm but cheaper.



Convergence Results

Theorem Let γk,m be the contraction number of the k-th levelmultigrid V-cycle, F-cycle or W-cycle algorithm with m pre-smoothing and m post-smoothing steps. Then

γk,m ≤ Cm−α

where the positive constant C is independent of k and m, pro-vided

m ≥ m∗

for a sufficiently large m∗ that is also independent of k.

(α = index of elliptic regularity)



Convergence Results


γk,m ≤ Cm−α

I W-cycle: Bank-Dupont 1981, B. 1999

I V-cycle, F-cycle: B. 2004 (additive multigrid theory)



Convergence Results


γk,m ≤ Cm−α

I W-cycle: Bank-Dupont 1981, B. 1999

I V-cycle, F-cycle: B. 2004 (additive multigrid theory)

I One of the key ingredients of the proofs is the equivalenceof a scale of discrete mesh-dependent norms (useful forthe smoothing property) and the scale of Sobolev norms(useful for the approximation property), which is estab-lished through the enriching operator.



Convergence Results


γk,m ≤ Cm−α

ReferenceB. and Sung

Multigrid algorithms for C0 interior penalty methods

SIAM J. Numer. Anal. (2006)



Convergence Results


γk,m ≤ Cm−α

This contraction number estimate is similar to the contractionnumber estimate for second order problems because B−1

k Ak

behaves like a second order differential operator.



Convergence Results


γk,m ≤ Cm−α

If we use a Richardson relaxation scheme without a precondi-tioner as the smoother, i.e.,

znew = zold + ωk(ψ − Akzold)

then we have a typical contraction number estimate for fourthorder problems

γk,m ≤ Cm−α/2

The effect of 100 smoothing steps without the preconditioneris (roughly) equivalent to the effect of 10 smoothing steps withthe preconditioner.



Numerical Results (Clamped Plates)

k = 2k = 1k = 0

Boundary conditions u = ∂u/∂n = 0

σ = 5

Vk(⊂ H10(Ω)) = Q2 rectangular finite element space

nk = dim(Vk) = (2k+1 − 1)2

κ(Ak) ≈ h−4k



Numerical Results (Clamped Plates)

k = 2k = 1k = 0

n0 = 1 κ0 = 1.0× 100

n1 = 9 κ1 = 4.0× 101

n2 = 49 κ2 = 1.6× 103

n3 = 225 κ3 = 2.9× 104

n4 = 961 κ4 = 4.8× 105

n5 = 3969 κ5 = 7.8× 106

n6 = 16129 κ6 = 1.3× 108

n7 = 65025 κ7 = 2.0× 109



Contraction Numbers in the Energy Norm

0.310.340.350.56

0.190.210.230.260.290.320.43

0.090.110.130.150.180.220.27

0.011 0.006 0.0032 0.00170.020.04

0.28

0.44

10

0.290.310.340.360.390.75

0.270.300.330.350.390.430.70

0.270.290.310.340.370.420.64

0.230.25

0.08

987654

7

6

5

4

3

2

1

mk

V-cycle Algorithm




0.26 0.24 0.22 0.19 0.17

0.51 0.41 0.37 0.34 0.31 0.28 0.26 0.24 0.22

0.30

0.002

0.72 0.49 0.24 0.27 0.22 0.18 0.15 0.13 0.11 0.09

0.71 0.51 0.40 0.34

0.53

0.29 0.27 0.25 0.23

0.80

0.76

0.38

0.38

0.42

0.42

0.82

0.83

0.53

0.53 0.32

0.42 0.38 0.34 0.31 0.29 0.26 0.24 0.23

0.34 0.32 0.29 0.26 0.25 0.22

0.34

0.003

4 5 6 7 8 9 10

0.53 0.28 0.15 0.08 0.04 0.02 0.01 0.006

3km

1

2

3

4

5

6

7

1 2

W-cycle Algorithm




0.240.260.280.310.340.370.42

0.180.190.220.240.270.300.34

0.00170.00320.0060

0.50 0.35 0.27 0.22 0.18 0.15 0.13 0.11 0.09

0.52 0.40

0.22

0.53

0.53

0.230.250.270.29

0.32

0.320.350.380.460.54

0.230.25

0.53 0.43 0.37 0.34 0.31 0.29 0.27 0.25 0.23

0.44 0.38 0.34 0.29 0.27

0.010.020.040.080.150.28

1098765432km

1

2

3

4

5

6

7

F-cycle Algorithm




0.68

0.63

0.54

0.42

0.46

0.06

0.73

0.70

0.64

0.57

0.64

0.46

0.06

0.760.80

0.06

0.47

0.47

0.60

0.69

0.74

0.78

0.06

0.46

0.42

0.58

0.66

0.72

0.71 0.56

0.56

0.52

0.49

0.63

0.45

0.05

0.61

0.60

0.56

0.51

0.36

0.45

0.050.06

0.46

0.40

0.52

0.61

0.65

0.68

0.05

0.46

0.41

0.50

0.57

0.62

0.65

0.76

75 76 77 78 79 80 81 82 83

0.06

0.47

0.64

0.60

0.71

7

km

1

2

3

4

5

6

V-cycle Algorithm Without a Preconditioner



Domain Decomposition Methods


Fast Solvers for Higher Order Problems Domain Decomposition Methods

Overlapping Domain Decomposition

Th TH Ωj

I given a triangulation Th

I create a coarse grid TH

I enlarge the subdomains of the coarse grid to constructan overlapping domain decomposition




Th TH Ωj

Vh ⊂ H10(Ω) = Qk (k ≥ 2) finite element space associated with

Th

VH ⊂ H10(Ω) = Q1 or Q2 finite element space associated with

TH

Vj ⊂ H10(Ωj) = Qk (k ≥ 2) finite element space associated with

Th (1 ≤ j ≤ J)




Ah : Vh −→ V ′h

〈Ahv1, v2〉 = Ah(v1, v2) ∀ v1, v2 ∈ Vh

Ah(w, v) =∑R∈Th

∫R

D2w : D2v dx

+∑e∈Eh

∫e

(∂2w∂n2

[[∂v∂n

]]+∂2v∂n2

[[∂w∂n

]])ds

+∑e∈Eh

σ

|e|

∫e

[[∂w∂n

]] [[∂v∂n

]]ds (clamped plates)

Ah is the SPD operator to be preconditioned.



Ingredients for the Preconditioner




A0 : VH −→ V ′H

〈A0v1, v2〉 = AH(v1, v2) ∀ v1, v2 ∈ VH

AH(w, v) =∑

R∈TH

∫R

D2w : D2v dx

+∑e∈EH

∫e

(∂2w∂n2

[[∂v∂n

]]+∂2v∂n2

[[∂w∂n

]])ds

+∑e∈EH

σ

|e|

∫e

[[∂w∂n

]] [[∂v∂n

]]ds




A0 : VH −→ V ′H

〈A0v1, v2〉 = AH(v1, v2) ∀ v1, v2 ∈ VH

I0 : VH −→ Vh

I0 = Πh EH

where EH : VH −→ H20(Ω) is the enriching operator from VH

to a Bogner-Fox-Schmit space associated with TH and Πh :H2

0(Ω) −→ Vh is the nodal interpolation operator.




A0 : VH −→ V ′H

〈A0v1, v2〉 = AH(v1, v2) ∀ v1, v2 ∈ VH

I0 : VH −→ Vh

I0 = Πh EH

where EH : VH −→ H20(Ω) is the enriching operator from VH

to a Bogner-Fox-Schmit space associated with TH and Πh :H2

0(Ω) −→ Vh is the nodal interpolation operator.

Remark The correct choice of the connection operator I0 iscrucial for the good performance of the preconditioner.




Aj : Vj −→ V ′j

〈Ajv1, v2〉 = Aj(v1, v2) ∀ v1, v2 ∈ Vj

Aj(w, v) =∑R∈Th

R⊂Ωj

∫R

D2w : D2v dx

+∑e∈Eh

e⊂Ωj

∫e

(∂2w∂n2

[[∂v∂n

]]+∂2v∂n2

[[∂w∂n

]])ds

+∑e∈Eh

e⊂Ωj

σ

|e|

∫e

[[∂w∂n

]] [[∂v∂n

]]ds




Aj : Vj −→ V ′j

〈Ajv1, v2〉 = Aj(v1, v2) ∀ v1, v2 ∈ Vj

Ij : Vj −→ Vh

Ij = natural injection



Two-Level Additive Schwarz Preconditioner

BTL : V ′h −→ Vh

BTL =J∑

j=0

IjA−1j It

j

Dryja and Widlund 1987

〈Itjα, vj〉 = 〈α, Ijvj〉 ∀α ∈ V ′, vj ∈ Vj



Two-Level Additive Schwarz Preconditioner

BTL : V ′h −→ Vh

BTL =J∑

j=0

IjA−1j It

j

Dryja and Widlund 1987

〈Itjα, vj〉 = 〈α, Ijvj〉 ∀α ∈ V ′, vj ∈ Vj

I The coarse grid solve A−10 and the subdomain solves A−1

j(1 ≤ j ≤ J) can be carried out in parallel.

I The coarse grid solve provides global communicationamong the subdomains so that this preconditioner is scal-able.



Condition Number Estimates

κ(BTLAh) =λmax(BTLAh)λmin(BTLAh)

≤ C(

1 +Hδ

)3

where δ measures the overlap among the subdomains andthe positive constant C is independent of h, H and J.




κ(BTLAh) =λmax(BTLAh)λmin(BTLAh)

≤ C(

1 +Hδ

)3

where δ measures the overlap among the subdomains andthe positive constant C is independent of h, H and J.

I κ(Ah) ≈ h−4

I κ(BTLAh) ≤ C if H/δ is bounded

(optimal preconditioner in the case of generous overlap)




I characterization of λmax(BTLAh)

λmax(BTLAh) = maxv∈Vh

〈Ahv, v〉

minv=

∑Jj=0 Ijvj

vj∈Vj

J∑j=0

〈Ajvj, vj〉

I characterization of λmin(BTLAh)

λmin(BTLAh) = minv∈Vh

〈Ahv, v〉

minv=

∑Jj=0 Ijvj

vj∈Vj

J∑j=0

〈Ajvj, vj〉




I The numerator in the two formulas satisfies the equiva-lence

〈Ahv, v〉 = Ah(v, v) ≈∑R∈Tk

|v|2H2(R) +∑e∈Eh

σ

|e|

∫e

[[∂v∂n

]]2ds






|v|2H2(R) +∑e∈Eh

σ

|e|

∫e

[[∂v∂n

]]2ds

I The term∑

R∈Tk|v|2H2(R)

also appears in classical noncon-forming methods for fourth order problems and can behandled by domain decomposition techniques for suchproblems.

B. 1996






|v|2H2(R) +∑e∈Eh

σ

|e|

∫e

[[∂v∂n

]]2ds

I The novelty is in controlling the sum∑e∈Eh

σ

|e|

∫e

[[∂v∂n

]]2ds.






|v|2H2(R) +∑e∈Eh

σ

|e|

∫e

[[∂v∂n

]]2ds


σ

|e|

∫e

[[∂v∂n

]]2ds.

The magnitudes of the scaling factor 1/|e| are different forthe fine and coarse meshes. If we just use the natural in-jection as I0, the condition number will depend adverselyon the ratio between the mesh sizes.






|v|2H2(R) +∑e∈Eh

σ

|e|

∫e

[[∂v∂n

]]2ds


σ

|e|

∫e

[[∂v∂n

]]2ds.

I We avoid this problem by including the enriching operatorin the definition of I0:

I0 = Πh EH

Because functions in the Bogner-Fox-Schmit spaces areC1 functions, the sum involving the jumps of the normalderivatives disappears.





ReferenceB. and Wang

Two-level additive Schwarz preconditioners for C0 interiorpenalty methods

Numer. Math. (2005)





ReferenceB. and Wang

Two-level additive Schwarz preconditioners for C0 interiorpenalty methods

Numer. Math. (2005)

I We can also consider nonoverlapping domain decom-position algorithms for C0 interior penalty methods.(work in progress)



Numerical Results

biharmonic equation with the boundary conditions ofclamped plates on the unit square

Vh = Q2 finite element space

VH = Q2/Q1 finite element space

σ = 5



Numerical Results

Number of PCG iterations needed for reducing the error in theenergy norm (not the residual error) by a factor of 10−6

13

-72

-62

-52

-52

-42

-42

-32

13

293131

16181818

1213131414

11121213

-32

-22

-22

hδ

4 subdomains, H = 1/2, Q2 coarse space



Numerical Results

Number of PCG iterations needed for reducing the error in theenergy norm by a factor of 10−6

2-3

2-4

2-4

2-5

2-5

2-7

16 16 16 15 14

15 14 14 13

22 20 19

-3

2-6δ

h

2




Numerical Results


2-5

-42

2-5

δ h2

-7

17 16 16 15

16 15 14

-42

-62




Numerical Results


6416

2

4

8

256

12 15 16 16

13 14 15 16

18 20 22

4H δJ

h = 2−6, Q2 coarse space

(scalable)



Numerical Results

1

1.56 10

2.81 10

4.62 10

1.12 10

x 2.30

x

x

x

-1

-1

-2

-3

10

10

101.69

1.04 1

64

32

16

8

4

x

102.04 x

x

x

x 4

3

2

4.30

10x -4102.954.59

4.66

4.75

4.80

4.83PSfrag replacements

H/δ λmax λmin κ(BTLAh)

4 subdomains, H = 1/2, h = 2−7

Q1 coarse space

κ(Ah) ≈ 2× 109 κ(BTLAh) ≤ C(

1 +Hδ

)3



Numerical Results

Number of PCG iterations needed for reducing the error(in the energy norm) by a factor of 10−6

2-3

2-4

2-4

2-5

2-5

2-7

19 19 17 17 15

22 21 20 18

41 39 35

-3

2-6δ

h

2


(with enriching operator)Susanne C. Brenner IMA Workshop Fast Solution Techniques


Numerical Results

Number of PCG iterations needed for reducing the error(in the energy norm) by a factor of 10−6

2-3

2-4

2-4

2-5

2-5

2-7

24 28 30 33 37

35 42 49 58

60 70 84

-3

2-6δ

h

2


(without enriching operator)Susanne C. Brenner IMA Workshop Fast Solution Techniques


Adaptive Methods


Fast Solvers for Higher Order Problems Adaptive Methods

Quadratic C0 IP Method for Clamped Plates


Ah(uh, v) =∫

Ωfv dx ∀v ∈ Vh

Ah(w, v) =∑T∈Th

∫T

D2w : D2v dx

+∑e∈Eh

∫e

∂2w∂n2

[[∂v∂n

]]ds +

∑e∈Eh

∫e

∂2v∂n2

[[∂w∂n

]]ds

+∑e∈Eh

σ

|e|

∫e

[[∂w∂n

]][[∂v∂n

]]ds

Vh ⊂ H10(Ω) is the P2 simplicial Lagrange finite element space.



A Residual Based Error Estimator

Definition

η2h =

∑T∈Th

η2T +

∑e∈Eh

η2e,1 +

∑e∈E i

h

η2e,2

Eh = the set of edgesE i

h = the set of interior edges




Definition

η2h =

∑T∈Th

η2T +

∑e∈Eh

η2e,1 +

∑e∈E i

h

η2e,2

residual associated with a triangle

ηT = h2T‖f‖L2(T)




Definition

η2h =

∑T∈Th

η2T +

∑e∈Eh

η2e,1 +

∑e∈E i

h

η2e,2

residual associated with the jump of the normal derivativeacross an edge

ηe,1 =σ

|e|1/2 ‖[[∂uh/∂n]]‖L2(e)

(no additional programming effort: already part of the pro-gramming of the C0 interior penalty method)




Definition

η2h =

∑T∈Th

η2T +

∑e∈Eh

η2e,1 +

∑e∈E i

h

η2e,2

residual associated with the jump of the second normalderivative across an edge

ηe,2 = |e|1/2‖[[∂2uh/∂n2]]‖L2(e)

(no additional programming effort: already part of the pro-gramming of the C0 interior penalty method)




Definition

η2h =

∑T∈Th

η2T +

∑e∈Eh

η2e,1 +

∑e∈E i

h

η2e,2

Explicit Form

ηh =( ∑

T∈Th

h4T‖f‖2

L2(T) +∑e∈Eh

σ2

|e|‖[[∂uh/∂n]]‖2

L2(e)

+∑e∈E i

h

|e|‖[[∂2uh/∂n2]]‖2L2(e)

)1/2



Reliability

The actual error is bounded by the error estimator.

‖u− uh‖h ≤ Cηh

The positive constant C depends only on the shape regularity of Th.

I∑

T∈Th|u− uh|2H2(T)

≤ 2 |u− Ehuh|2H2(Ω) +2∑

T∈Th|uh − Ehuh|2H2(T)

I∑


≤ C∑

e∈Eh

1|e|‖[[∂uh/∂n]]‖2

L2(e) ≤ C∑

e∈Ehη2

e,1



Reliability



recall‖v‖2

h =∑T∈Th

|v|2H2(T) +∑e∈Eh

σ

|e|‖[[∂v/∂n]]‖2

L2(e)

I∑


≤ 2 |u− Ehuh|2H2(Ω)+2∑


I∑


≤ C∑

e∈Eh1|e|‖[[∂uh/∂n]]‖2

L2(e) ≤ C∑

e∈Ehη2

e,1



Reliability



recall‖v‖2

h =∑T∈Th

|v|2H2(T) +∑e∈Eh

σ

|e|‖[[∂v/∂n]]‖2

L2(e)

I∑

e∈Ehσ|e|‖[[∂(u− uh)/∂n]]‖2

L2(e) =∑

e∈Ehσ|e|‖[[∂uh/∂n]]‖2

L2(e)

= σ−1∑e∈Eh

η2e,1

I∑


≤ 2 |u− Ehuh|2H2(Ω)+2∑


I∑


≤ C∑

e∈Eh1|e|‖[[∂uh/∂n]]‖2

L2(e) ≤ C∑

e∈Ehη2

e,1

ηe,1 = σ|e|1/2 ‖[[∂uh/∂n]]‖L2(e)



Reliability



recall‖v‖2

h =∑T∈Th

|v|2H2(T) +∑e∈Eh

σ

|e|‖[[∂v/∂n]]‖2

L2(e)

I∑


≤ 2 |u− Ehuh|2H2(Ω)+2∑


I∑


≤ C∑

e∈Eh1|e|‖[[∂uh/∂n]]‖2

L2(e) ≤ C∑

e∈Ehη2

e,1

Eh : Vh −→ H20(Ω) (enriching operator)



Reliability



recall‖v‖2

h =∑T∈Th

|v|2H2(T) +∑e∈Eh

σ

|e|‖[[∂v/∂n]]‖2

L2(e)

I∑


≤ 2 |u− Ehuh|2H2(Ω)+2∑


I∑


≤ C∑

e∈Eh1|e|‖[[∂uh/∂n]]‖2

L2(e) ≤ C∑

e∈Ehη2

e,1



Reliability



recall‖v‖2

h =∑T∈Th

|v|2H2(T) +∑e∈Eh

σ

|e|‖[[∂v/∂n]]‖2

L2(e)

I∑


≤ 2 |u− Ehuh|2H2(Ω)+2∑


I∑


≤ C∑

e∈Eh1|e|‖[[∂uh/∂n]]‖2

L2(e) ≤ C∑

e∈Ehη2

e,1



Reliability



|u− Ehuh|H2(Ω)

= supζ∈H2

0(Ω)\0

(D2(u− Ehuh),D2ζ)L2(Ω)

|ζ|H2(Ω)

≤ C( ∑

T∈Th

η2T +

∑e∈Eh

η2e,1 +

∑e∈E i

h

η2e,2

)1/2

ηT = h2T‖f‖L2(T) ηe,1 = σ|e|−1/2‖[[∂uh/∂n]]‖L2(e)

ηe,2 = |e|1/2‖[[∂2uh/∂n2]]‖L2(e)



Efficiency

The error estimator is bounded by actual error plus dataoscillation.

ηh ≤ C(σ‖u− uh‖2

h +∑T∈Th

h4T‖f − f‖2

L2(T)

)1/2

wheref∣∣T =

1|T|

∫T

f dx ∀T ∈ Th



Efficiency

The error estimator is bounded by actual error plus dataoscillation.

ηh ≤ C(σ‖u− uh‖2

h +∑T∈Th

h4T‖f − f‖2

L2(T)

)1/2

wheref∣∣T =

1|T|

∫T

f dx ∀T ∈ Th

I usual bubble function techniques (with some twists)

I part of the medius analysis



Efficiency

Details can be found in the following paper.

ReferenceB., Gudi and Sung

An a posteriori error estimator for a quadratic C0 interiorpenalty method for the biharmonic problem

IMA J. Numer. Anal. (2010)



Numerical Results

We have implemented an adaptive algorithm using the er-ror estimator ηh and the bulk marking strategy of Dörfler, andtested it on an L-shaped domain using an exact solution withthe correct singularity.



Error and Error Estimator on Adaptive Meshes

103

104

105

100

101

Degrees of Freedom

Err

or

and

Est

imat

or

Error Estimator



Comparison of Uniform and Adaptive Meshes

103

104

105

100

101

Degrees of Freedom

Err

or

and

Est

imat

or

Error on Uniform MeshError on Adaptive Mesh



Concluding Remarks


Fast Solvers for Higher Order Problems Concluding Remarks

I For higher order problems discontinuous Galerkin meth-ods are simpler in many aspects than standard finite ele-ment methods and fast solvers (multigrid, domain decom-position, adaptive) are available.




I We consider C0 interior penalty methods for fourth orderproblems because they involve fewer unknowns and theyuse standard finite element spaces for second order prob-lems which already exist in many software packages. Butone can also consider totally discontinuous finite elementspaces in order to handle hanging nodes (h-adaptivity)and/or p-adaptivity.




I We consider C0 interior penalty methods for fourth orderproblems because they involve fewer unknowns and theyuse standard finite element spaces for second order prob-lems which already exist in many software packages. Butone can also consider totally discontinuous finite elementspaces in order to handle hanging nodes (h-adaptivity)and/or p-adaptivity.

I Totally discontinuous interior penalty methods for ellipticequations of arbitrary order on smooth domains were al-ready considered by Baker in 1977. For fourth order prob-lems on polygonal domains, such methods were studiedby Süli et al. in several recent papers.



I Since the finite element spaces for discontinuousGalerkin methods for elliptic problems of order 2` can alsobe used for elliptic problems of lower order, we can de-velop multigrid algorithms recursively through the hierar-chy of elliptic problems, i.e., multigrid methods for secondorder problems can be embedded in the smoothers formultigrid methods for fourth order problems, which canthen be embedded in multigrid methods for sixth orderproblems, etc.




I The asymptotic rate of decrease for the contraction num-bers of such embedded multigrid methods will behave likem−α, instead of m−α/` (m = number of smoothing steps,α = index of elliptic regularity). The complexity of theseembedded multigrid methods remains optimal.




I The asymptotic rate of decrease for the contraction num-bers of such embedded multigrid methods will behave likem−α, instead of m−α/` (m = number of smoothing steps,α = index of elliptic regularity). The complexity of theseembedded multigrid methods remains optimal.

I The convergence analysis of these embedded multigridmethods can be handled by the additive multigrid theory.



I We can retain the good performance of domain decom-position methods by using enriching operators to connectdiscontinuous finite element spaces to conforming finiteelement spaces.




I Fast solvers for higher order problems in 3D can be de-veloped. But the convergence analysis of multigrid meth-ods is more challenging because the elliptic regularity fornon-smooth domains in 3D is more complicated.





I We can combine multigrid and domain decomposition byusing multigrid to solve the subdomain problems in paral-lel.





I We can combine multigrid and domain decomposition byusing multigrid to solve the subdomain problems in paral-lel.

I The convergence of adaptive algorithms for discontinu-ous Galerkin methods for higher order problems remainsan open problem.



Acknowledgement


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