Convergence of goal-oriented adaptive finite element ... · Thesis defense Sara Pollock Thesis advisor: Michael Holst Collaborators: Michael Holst, Yunrong Zhu UCSD Mathematics May

Convergence of goal-oriented adaptive finite elementmethods

Thesis defense

Sara Pollock

Thesis advisor: Michael Holst

Collaborators: Michael Holst, Yunrong Zhu

UCSD Mathematics

May 9, 2012

S. Pollock 1/56 Convergence of GOAFEM

UCSD Mathematics Thesis defense, May 9, 2012

Outline1 Introduction

Research planAdaptive and goal-oriented methodsFunction spacesContraction frameworkContraction theorems

2 The linear problemThe primalThe dualConvergence of the quantity of interest

3 The nonlinear problemThe primalThe dualNew convergence estimatesContraction estimatesConvergence of the quantity of interest

4 Numerics and complexityNumerical results (nonlinear problem)Complexity: linear problem

5 Future work6 ReferencesS. Pollock 2/56 Convergence of GOAFEM


What we said we would do

The Research plan from my candidacy talkContraction result for goal oriented method applied to the elliptic problem usingframework including data-oscillation as described in Candidacy talk.

I Convergence established for idealized version (piecewise polynomial data).I Complexity results pending.I Contraction for practical method (L2 data) to follow.

Contraction result for goal oriented method applied to the elliptic problem usingframework discussed in [Holst, Tsogtgerel, and Zhu, 2009].

Contraction result for goal oriented method applied to the semilinear problemusing framework discussed in [Holst, Tsogtgerel, and Zhu, 2009].

Extend our results to the semilinear problem with nonsymmetric linear part.



What we did

The Research plan we completed

• Contraction result for goal-oriented method applied to the elliptic problem usingframework including data-oscillation as described in Candidacy talk.

• Convergence established for idealized version (piecewise polynomial data). Complexity results pending. Contraction for practical method (L2 data) to follow.

√Contraction and complexity results for goal-oriented method applied to theelliptic problem based on frameworks discussed in[Holst, Tsogtgerel, and Zhu, 2009] and[Cascon, Kreuzer, Nochetto, Siebert, 2008].

√Contraction result for goal-oriented method applied to the semilinear problemextending framework discussed in [Holst, Tsogtgerel, and Zhu, 2009].

Extend our results to the semilinear problem with nonsymmetric linear part.



Adaptive methods

Adaptive finite element methods (AFEM) are those in which only select elements arerefined at each iteration of the algorithm. In contrast, uniform methods globally refinethe mesh at every step.

Adaptive methods are effective at reducing the overall complexity or degrees offreedom in the problem and are of particular interest in problems with localizedsingularities. In this work, we are interested in extending convergence theory forAFEM, specifically for a goal-oriented AFEM (GOAFEM).

In an adaptive method, the error is estimated on each element by means of acomputable error estimator (sometimes called error indicator) then certain elementsare selected for refinement.

We focus our work on residual-based estimators.



Motivation 1: contraction for adaptive methods

1 [Mekchay, Nochetto, 2005] Contraction for (nonsymmetric) elliptic problem. Theform of error that contracts: "total error"

|||u−uk |||2 + γosc 2k (uk ).

2 [Cascon, Kreuzer, Nochetto, Siebert, 2008] Contraction for (symmetric) ellipticproblems and [Holst, Tsogtgerel, and Zhu, 2009] contraction for nonlinear,including semilinear problems. The form of error that contracts: "quasi-error"

|||u−uk |||2 + γη2k (uk ).

3 [Bank, Holst, Szypowski, and Zhu, 2011] Contraction for semilinear problem withinexact solvers. The form of error that contracts:

|||u−uk |||2 + γη2k (uk )

where uk is the inexact Galerkin solution.



Goal oriented methods: the idea

In a goal-oriented method, we are interested in a functional (or more generally afunction) of the weak solution rather than the solution itself.

This quantity g(u) is often called the quantity of interest.

For example, the function g(·) may be a (weighted) average over a subdomain or aline integral about its boundary. Goal-oriented methods may be used to estimate aphysical quantity, or be used in pointwise a posteriori error estimation, generally usingmollification.

For lots of examples of applications particularly to CFD, see [Giles, Süli, 02] or forstructural examples see [Grätsch, Bathe, 2005].

Goal-oriented methods involve solving a second PDE called the dual whose solution is(sometimes) called the influence function.



Goal oriented methods: the residual and the dual

Let R(uh) the residual of the original or primal problem (more on this later). Then thedual problem is chosen so the dual solution z satisfies the relation

g(u−uh) = 〈R(uh),z〉= 〈R(uh),z− vh〉, for all vh ∈ Vh.

In dual weighted residual (DWR) methods, z is an approximation to z solved in a moreenriched space than Vh, and for πh a projector onto vh yielding

g(u−uh)≈ 〈R(uh), z−πhz〉

where the RHS is a computable quantity and can be used as an approximation of theerror or can be taken elementwise and used as an error indicator to drive an adaptivemethod.

On the other hand, we can show

g(u−uh) = 〈R(uh),z− vh〉 ≤ K(|||u−uh|||2 + |||z− zh|||2

)and determining contraction of a quantity that bounds the RHS we can analyticallydetermine convergence of the method. We do it this way.



DWR: g(u−uh)≈ 〈R(uh), z−πhz〉

Most of the literature on GOAFEM focus on the DWR technique where the estimatorrelies on an approximate solution to the dual problem to drive the refinement. See, forexample, [Prudhomme, Oden, 1999], [Estep, Holst, Larson, 2002],[Grätsch, Bathe, 2005], [Carey, Estep, Tavener, 2009].

These examples are all well-supported by numerics, (not necessarily with DWRrefinement alone); however, they do not show strong contraction or a monotonicdecrease in any form of the error.

For linear problems, there is an exact dual problem so z ≈ z is reasonable and z−πhzis a meaningful weight for the residual. In nonlinear problems, the dual problem mustbe approximated as a function of uh.

As discussed in [Dahmen, Kunoth, Vorloeper, 2006], it is not clear that g(uh)→ g(u),even for linear problems.

Exception: Convergence is shown for a DWR GOAFEM for the Laplacian assumingu,z ∈ C3(Ω), on adaptive hanging node meshes of parallel squares[Moon, Schwerin, Szepessy, Tempone, 2006].



Goal oriented methods: contraction!

In [Mommer, Stevenson, 2009] it was shown for the scaled Laplacian that solving adual problem together with the original PDE can help drive the adaptive methodtowards a solution g(u) faster than solving the original problem for the solution ualone. They also show contraction of the error at each iteration.

|g(u)−g(uk )| ≤ |||u−uk ||||||z− zk |||

with contraction shown for each of |||u−uk |||2 and |||z− zk |||2. They also show thecomplexity result that given ε > 0 then method solves to the accuracy

|||u−uk ||||||z− zk ||| ≤ ε

where the cardinality of the mesh satisfies

#Tk −#T0 = O(

ε−1/(s+t)

)where s, t > 0 depend on the approximation classes that u and z belong to.

We base our goal-oriented method on their ideas.



In defense of the energy norm

One of the motivations for goal-oriented error estimation is that the global energy-normestimate may not give relevant information [Prudhomme, Oden, 1999] and the energynorm overestimates the error in the quantity of interest [Grätsch, Bathe, 2005],[Estep, Holst, Larson, 2002].

In terms of the contraction framework, the energy norm error is actually the quantitythat gets reduced between consecutive iterations. In the symmetric case, orthogonalityin the energy norm

|||u−u2|||2 = |||u−u1|||2−|||u1−u2|||2

or in the more general case: "quasi-orthogonality"

|||u−u2|||2 ≤ Λ|||u−u1|||2−|||u1−u2|||2, Λ > 1 but Λ≈ 1.

In our framework, the energy norm is the quantity that effectively gets smaller at eachiteration!



Our results

1 [Holst, Pollock, 2011] Convergence of goal-oriented method for nonsymmetricelliptic problems

|g(u)−g(uk )| ≤ 2|||u−uk ||||||z− zk ||| ≤ |||u−uk |||2 + |||z− zk |||2

with contraction shown for each of the quasi-errors

|||u−uk |||2 + γp η2k (uk ) and |||z− zk |||2 + γd ζ

2k (zk ).

Plus, we show quasi-optimal complexity.2 [Holst, Pollock, Zhu, 2012] Convergence of goal-oriented method for semilinear

problems|g(u)−g(uk )| ≤ C

(|||u−uk |||2 + |||z− zk |||2

)with contraction shown for the combined quasi-error

|||z− zk |||2 + γζ2k (zk ) + τ |||u−uk |||2 + τγpη

2k (uk ).



Problem setup: Sobolev space

In the weak formulations of our problems, we seek solutions u and consider testfunctions v in the Sobolev space H1

0 (Ω).The native norm is the Sobolev H1 norm given by

‖v‖2H1 = 〈∇v ,∇v〉+ 〈v ,v〉.

The Sobolev space H10 (Ω) is then a subset of L2(Ω) and can be defined

H10 (Ω) =

u ∈ L2(Ω)

∣∣∣ ‖u‖H1(Ω) < ∞ and u∣∣∂Ω

= 0

where the restriction to the boundary is meant in the sense of the trace. More precisely,

H10 (Ω) = C∞

c (Ω)‖ · ‖H1

.

Fact: H10 (Ω) and L2(Ω) are both Hilbert spaces.



The mesh and the finite element spaces

We consider approximating the solution to our PDE in the sequence of nested spaces

V0 ⊆ V1 ⊆ V2 ⊆ . . .⊆ V = H10 (Ω).

We employ a standard conforming piecewise polynomial finite element approximationbelow.We make the following assumptions on the underlying simplex mesh:

1 The initial mesh T0 is conforming.2 The mesh is refined by newest vertex bisection [Binev, Dahmen, DeVore, 2004],

[Mommer, Stevenson, 2009] at each iteration.3 The initial mesh T0 is sufficiently fine (details to follow).

Define the finite element space

VT := H10 (Ω)∩ ∏

T∈TPn(T ) and Vk := VTk .

where Pn(T ) is the space of polynomials degree degree n over T .

Notation: the mesh diameter

hT := maxT∈T

hT , where hT = |T |1/d .



The adaptive loop

The goal oriented adaptive finite element method (GOAFEM) is based on the standardAFEM algorithm:

SOLVE → ESTIMATE → MARK → REFINE .

Solve: In practice the linear problem may be solved by a standard iterative solver. Thenonlinear problem may be solved by a standard inexact Newton + multilevel algorithm.

Here we assume the exact solution to each problem.



Estimate!


We use an error indicator derived from the strong-form of the residual. The errorindicator is given elementwise as

η2T (v ,T ) := h2

T‖R(v)‖2L2(T ) + hT‖JT (v)‖2

L2(∂T ), v ∈ VT

The dual residuals are defined analogously. The jump residual for primal and dualproblems in all cases is

JT (v) := J[A∇v ] ·nK∂T

where jump operator J · K is given by

JφK∂T := limt→0

φ(x + tn)−φ(x− tn)

and n is taken to be the appropriate outward normal defined piecewise on ∂T . Theerror estimator is given by the l2 sum of indicators. For the Galerkin solution uk we usethe notation

η2k = ∑

T∈Tk

η2T (uk ,T ).




The mesh is marked at each iteration using the Dörfler stategy with respect to bothprimal and approximate dual problems: Given θ ∈ (0,1)

Mark a set Mp ⊂ Tk such that,

∑T∈Mp

η2k (uk ,T )≥ θ

2η

2k (uk ,Tk )

Mark a set Md ⊂ Tk such that,

∑T∈Md

ζ2k (zk ,T )≥ θ

2ζ

2k (zk ,Tk )

Let M = Mp ∪Md the union of sets found for the primal and dual problemsrespectively.

Refine: The refinement (including the completion) is performed according to newestvertex bisection.



3 ingredients for contraction

As in [Holst, Tsogtgerel, and Zhu, 2009] and[Cascon, Kreuzer, Nochetto, Siebert, 2008], the contraction argument for the primaland dual follows from first establishing three preliminary results for two successiveAFEM approximations u1 and u2.

1 Quasi-orthogonality: There exists ΛG > 1 such that

|||u−u2|||2 ≤ ΛG|||u−u1|||2−|||u2−u1|||2.2 Error estimator as upper bound on error: There exists C1 > 0 such that

|||u−uk |||2 ≤ C1η2k (uk ,Tk ), k = 1,2.

3 Estimator reduction: For M the marked set that takes refinement T1→ T2, forpositive constants λ < 1 and Λ1 and any δ > 0

η22(v2,T2)≤ (1 + δ)η2

1(v1,T1)−λη21(v1,M )+ (1 + δ

−1)Λ1η20|||v2− v1|||.

Together with the Dörfler marking strategy, these three estimates are combined toshow the contraction of the (primal) quasi-error

|||u−u2|||2 + γp η2k (u2)≤ α

2|||u−u1|||2 + γp η2k (u1)

, α ∈ (0,1).



For comparison: 4-ingredient contraction

Another related method for showing contraction involves the first two of theseestimates, and two others which estimator reduction.

This framework is used to show contraction in [Mekchay, Nochetto, 2005]. The methodin [Mommer, Stevenson, 2009] closely resembles this as well. Due to the simplicity ofthat problem, the oscillation terms don’t show up in their initial estimates. One of thedisadvantages of this method is that it assumes an "interior node property:" Everyelement in the refinement set contains a vertex of the new refinement in its interior.This method also requires marking the mesh for both indicators and oscillation.

Estimator as lower bound on energy norm error:

c2η(u1)≤ |||u−u1|||+ d2osc 1(u1).

Oscillation control: there are constants 0 < 1 < ρ1 and ρ2 > 0 with

osc 22(u2)≤ ρ1osc 2

1(u1) + ρ2|||u2−u1|||2.

The oscillation term consists of the high-frequency components of the residual. We dorequire the lower bound estimate for the complexity analysis.










Nonsymmetric linear problem

Our first results are for the nonsymmetric linear elliptic problem.

The PDE in strong form:

L(u) :=−∇ · (A∇u) + b ·∇u + cu = f , in Ω,

u = 0, on ∂Ω,

with weak formulation: Find u ∈ H10 (Ω) such that

a(u,v) :=Z

ΩA∇u ·∇v + b ·∇uv + cuv dx = f (v), ∀v ∈ H1

0 (Ω)

where we follow the convention of [Ciarlet, 2002] and associate f ∈ L2(Ω) with itsRiesz-representer

f (v) =Z

Ωfv dx .

The goal: Find g(u) for a given g ∈ L2(Ω).



Assumptions on the data

Let Ω⊂ Rd a polyhedral domain, d = 2 or 3.

The problem data D = (A,b, f ) and goal function g satisfy1 A : Ω→ Rd×d , Lipschitz, and a.e. symmetric positive-definite:

ess inf x∈Ωλmin(A(x)) = µ0 > 0,

ess sup x∈Ωλmax(A(x)) = µ1 < ∞.

2 b : Ω→ Rd , with bk ∈ L∞(Ω) , and b divergence-free.3 c : Ω→ R, with c ∈ L∞(Ω), and c(x)≥ 0 for all x ∈ Ω.4 f ,g ∈ L2(Ω).



The energy norm

Many of the a posteriori estimates are developed with respect to the energy norm.Define

|||v |||2 := a(v ,v).

This norm is seen to be induced by the symmetric part of the problem. The energynorm is seen to be equivalent to the native norm. In particular, there is a continuityconstant ME with

a(u,v)≤ME‖u‖H1‖v‖H1 , for all u,v ∈ H10 (Ω)

and a coercivity constant mE with

a(v ,v)≥m2E‖v‖2

H1 , for all v ∈ H10 (Ω)

yielding

m2E‖v‖2

H1 ≤ a(v ,v)≤ME‖v‖2H1 .

Continuity and coercivity imply existence and uniqueness of the solution by theLax-Milgram Theorem.



The dual problem

The dual problem is introduced to satisfy the relationship

g(u−uk ) = 〈R(uk ),z〉

The residual is given by

R(v) = f −L(v) = f + ∇ · (A∇u)−b ·∇u− cu

The formal adjoint of a( · , ·) satisfies the necessary relationship. Integrating by partson the convergence term

a(u,v) := 〈A∇u,∇v〉+ 〈b ·∇u,v〉+ 〈cu,v〉= 〈A∇v ,∇u〉−〈b ·∇v ,u〉+ 〈cv ,u〉=: a∗(v ,u).

The dual problem is defined: Find z ∈ H10 (Ω) such that

a∗(z,v) = g(v), for all v ∈ H10 (Ω)

Then

g(ek ) = a∗(z,u)−a∗(z,uk ) = a(u,z)−a(uk ,z) = f (z)−a(uk ,z) = 〈R(uk ),z〉.



Contraction

Theorem (Contraction of the quasi-error)Let θ ∈ (0,1], and let Tk ,Vkk≥0 be the sequence of meshes and finite elementspaces produced by GOAFEM. Then there are constants γp,γd > 0 and 0 < α < 1,depending on the initial mesh T0 and marking parameter θ such that

|||u−uk+1|||2 + γp η2k+1 ≤ α

2 (|||u−uk |||2 + γp η2k

),

and|||z− zk+1|||2 + γd ζ

2k+1 ≤ α

2 (|||z− zk |||2 + γd ζ2k

).

Contraction in the dual follows from contraction in the primal as the differentialoperators are the same up to the sign of the convection term and both f ,g ∈ L2(Ω).

The contraction proof for both primal and dual problems is similar to thatin [Cascon, Kreuzer, Nochetto, Siebert, 2008], where we used quasi-orthogonalitywherever they used orthogonality .



Convergence

Corollary (Bounding the error in the goal function)

Let the quasi-error for primal and dual problems given by

Q2k (uk ) = |||u−uk |||2 + γp η

2k

andQ2

k (zk ) = |||z− zk |||2 + γd ζ2k .

Then there is a constant α ∈ (0,1) with

|g(u)−g(uk )| ≤ Q2k (uk ) + Q2

k (zk )≤ α2k (Q2

0(u0) + Q20(z0)

).

The second inequality follows from iterating the previous theorem (contraction of thequasi-error).



Sketch of proof

By linearity and (primal) Galerkin orthogonality

g(u)−g(uk ) = a(u−uk ,z− zk ).

Now use the energy norm to bound the RHS. Separating out the symmetric andnonsymmetric parts

a(u−uk ,z− zk )≤ |||u−uk ||||||z− zk |||+ 〈b ·∇(u−uk ),z− zk 〉.

Applying L2-lifting (Aubin-Nitsche), bound the nonsymmetric part by

〈b ·∇(u−uk ),z− zk 〉 ≤ Λ|||z− zk ||||||u−uk |||

with Λ = ‖b‖L∞C∗hs

0µ−1/20 < 1 (mesh assumption) yielding

|g(u)−g(uk )|= |a(u−uk ,z− zk )| ≤ 2|||u−uk ||||||z− zk |||.

Putting this together with the contraction of the primal and dual quasi-errors

|g(u)−g(uk )|+ γpη2k + γd ζ

2k ≤ α

2k (|||u−u0|||2 + γpη20(u0,T0)

+|||z− z0|||2 + γd ζ20(z0,T0)

).










The PDE and the quantity of interest

Next we consider a semilinear problem.

The PDE in strong form:

N (u) :=−∇ · (A∇u) + b(u) = f , in Ω,

u = 0, on ∂Ω,

with the residualR(u) = f −N (u).

The weak form: Find u ∈ H10 (Ω) such that

a(u,v) + 〈b(u),v〉= f (v), ∀v ∈ H10 (Ω),

where the bilinear forma(u,v) =

ZΩ

A∇u ·∇v dx .

The goal: Find g(u) for a given g ∈ L2(Ω).



The data

Let Ω⊂ Rd a polyhedral domain, d = 2 or 3.The problem data D = (A,b, f ) and goal function g satisfy

1 A : Ω→ Rd×d , Lipschitz, and a.e. symmetric positive-definite:

ess inf x∈Ωλmin(A(x)) = µ0 > 0,

ess sup x∈Ωλmax(A(x)) = µ1 < ∞.

2 b : Ω×R→ R is smooth in the second argument. For simplicity, we write b(u)instead of b(x ,u). Moreover, we assume that b is monotone (increasing):

b′(ξ)≥ 0, for all ξ ∈ R.

3 f ,g ∈ L2(Ω).

remark: We say the problem is semilinear because the diffusion term is linear in uand b may be a nonlinear function of u, but not its derivatives.



Norm equivalence

Define the energy semi-norm by the principal part of the differential operator

|||v |||2 := a(v ,v).

Continuity of a( · , ·) follows from the Hölder inequality, and bounding the L2 norm ofthe function and its gradient by the H1 norm

a(u,v)≤ µ1‖u‖H1‖v‖H1 = ME‖u‖H1‖v‖H1 .

Non-negativity follows from the Poincaré inequality with constant CΩ

a(v ,v)≥ µ0|v |2H1 ≥ CΩµ0‖v‖2H1 = m2

E‖v‖2H1 ,



The linearized dual operator

We form the dual by linearization to satisfy the relation g(u−uk ) = 〈R(uk ),z〉. By theintegral mean-value theorem (generalized Taylor expansion)

b(u)−b(uk ) =Z 1

0b′(uk + t(u−uk ))dt(u−uk ) =

Z 1

0b′(tu + (1− t)uk )dt(u−uk )

yielding

b(u)−b(uk ) = Bk (u−uk ), Bk :=Z 1

0b′(tu + (1− t)uk )dt.

Noting Bk = B∗k we define the dual problem: Find z ∈ H10 (Ω) such that

a(z,v) + 〈Bk z,v〉= g(v), for all v ∈ H10 (Ω).

Then we have

g(ek ) = a(z,u−uk ) + 〈Bk z,u−uk 〉= a(z,u−uk ) + 〈z,Bk (u−uk )〉= a(z,u−uk ) + 〈z,b(u)−b(uk )〉= f (z)− (a(uk ,z) + 〈b(uk ),z〉) = 〈R(uk ),z〉.



The approximate and limiting duals

The linearized dual operator: Bk :=R 1

0 b′(tu + (1− t)uk ) dt =R 1

0 b′(uk + t(u−uk )).

Unfortunately, this operator is a function of the exact solution u so we can’t compute it.

In order to introduce a computable dual operator, one that is not a function of the exactsolution u, we define the approximate dual operator

b′(uk ).

This operator is instrumental for defining a computable a posteriori error indicator forthe dual problem.

A second main difficulty with Bk is it changes at each iteration, as does b′(uk ), aproblem for the contraction analysis. So, we introduce the limiting dual operator

b′(u).

This is again a function of the exact solution u and not computable.

However, our final contraction result is written in terms of this limiting operator. Ourcontraction and convergence arguments must now convert between limiting,approximate and linearized dual problems.



Summary: the continuous problems

Primal: Find u ∈ H10 (Ω) such that

a(u,v) + 〈b(u),v〉= f (v), ∀v ∈ H10 (Ω).

The limiting dual problem: Find z ∈ H10 (Ω) such that

a(z,v) + 〈b′(u)z,v〉= g(v), ∀v ∈ H10 (Ω).

Existence and uniqueness of solutions to the primal problem follows from standardvariational or fixed-point arguments. For the dual problems (all of them... they’re alllinear with non-negative reaction coefficients) the result may be derived from theLax-Milgram Theorem.



Discrete problems

We compute solutions to the following two problems:

Define the discrete primal problem: Find uk ∈ Vk such that

a(uk ,vk ) + 〈b(uk ),vk 〉= f (vk ), vk ∈ Vk .

The approximate dual problem (sequence): Find z jk ∈ Vk such that

a(z jk ,vk ) + 〈b′(uj )z j

k ,vk 〉= g(vk ) for all vk ∈ Vk .

We can’t compute this one but it is used in the analysis

The discrete limiting dual problem is given by: Find zk ∈ Vk such that

a(zk ,vk ) + 〈b′(u)zk ,vk 〉= g(vk ) for all vk ∈ Vk .



A priori estimates

We require the solutions to the primal and limiting and approximate dual problemssatisfy L∞ bounds. Such bounds have been established assuming various additionalconditions on either the growth of the nonlinearity b or on the angles of the mesh (see[Bank, Holst, Szypowski, and Zhu, 2011]).

We assume the following L∞ bounds on the primal and discrete primal solutions.There are u−,u+ ∈ L∞ which satisfy

u−(x) < u(x),uk (x)≤ u+(x) for almost every x ∈ Ω.

Then the following properties hold:1 b is Lipschitz on [u−,u+]∩H1

0 (Ω) for a.e. x ∈ Ω with constant B.2 b′ is Lipschitz on [u−,u+]∩H1

0 (Ω) for a.e. x ∈ Ω with constant Θ.3 There are z−,z+ ∈ L∞ which satisfy

z−(x) < z(x), zj (x), z jj (x)≤ z+(x) for almost every x ∈ Ω, j ∈ N

and there is a constant KZ := max‖z−‖L∞,‖z+‖L∞

.



Contraction overview

For the contraction of the combined quasi-error we require the same estimates asbefore, plus a few more. In particular for the limiting dual problem

1 Quasi-orthogonality: There exists ΛG > 1 such that

|||z− z2|||2 ≤ ΛG|||z− z1|||2−|||z2− z1|||2.

2 Error estimator as upper bound on error: There exists C1 > 0 such that

|||z− zk |||2 ≤ C1ζ2k (zk ,Tk ), k = 1,2.

3 Estimator reduction: For M the marked set that takes refinement T1→ T2, forpositive constants λ < 1 and Λ1 and any δ > 0

ζ22(v2,T2)≤ (1 + δ)ζ2

1(v1,T1)−λζ21(v1,M + (1 + δ

−1)Λ1η20|||v2− v1|||.

We also use the contraction of the primal problem

|||u−u2|||2 + γη2k (u2)≤ α

2|||u−u1|||2 + γη2k (u1)

, α ∈ (0,1).

And a few new estimates to handle switching between approximate and limitingproblems.



Switching between estimators

1 Dual perturbation over sets: for δ1,δA > 0

ζ21(v ,M1)≥ (1 + δ1)−1(1 + δA)−1

ζ21,1(w ,M1)

− (1 + δ1)−1δ−1A Θ2K 2

Z h20‖u−u1‖2

L2−δ−11 Λ2

1(d + 2)η20‖v−w‖2

H1

The first bounds the difference between the limiting and approximate estimatorswhich are functions of different residual operators.

2 Applying the Dörfler property to the limiting estimator

−ζ21(z1,M )≤− βθ2

(1 + δ4)ζ

21(z1,T1)− (1−β)θ2

(1 + δ4)|||z− z1|||2

+

(θ2

(1 + δA)(1 + δ2)δB+

1δ A

)Θ2K 2

Z C2∗h

2(1+s)0

(1 + δ1)|||u−u1|||2

+

(θ2

(1 + δ1)(1 + δA)δ2+

1δ 1

)Λ1η

20(D,T0)|||z1− z1

1 |||2.

The second follows from the first and both are mainly bookkeeping argumentsand follow from the Local Lipschitz property of the error estimator as establishedon the way to estimator reduction in both the primal and limiting dual problems.



We can handle that last term

Lemma (Bounding the error in the discrete problem)

|||z1− z11 ||| ≤ΘKZ C∗C∗h

2s0 |||u−u1|||.

The error between the Galerkin solutions to the limiting and approximate dualproblems is bounded by the error in the primal problem.

Of all the types of error between approximate, linearized and limiting dual problems,the error between discrete solutions is the most convenient, as z1− z1

1 ∈ V1.



Sketch of proof

Recall that

z1 solves a(z1,v) + 〈b′(u)z1,v〉= g(v), for all v ∈ V1 (1)

z11 solves a(z1

1 ,v) + 〈b′(u1)z11 ,v〉= g(v), for all v ∈ V1. (2)

Subtracting (2) from (1) and rearranging terms

a(z1− z11 ,v) + 〈(b′(u)−b′(u1))z1,v〉= 〈b′(u1)(z1

1 − z1),v〉, v ∈ V1. (3)

In particular, for v = z1− z11 ∈ V1 equation (3) yields

|||z1− z11 |||2 =−〈(b′(u)−b′(u1))z1, z1− z1

1〉−〈b′(u1)(z1− z11 ), z1− z1

1〉≤ −〈(b′(u)−b′(u1))z1, z1− z1

1〉 (4)

where the last line in (4) follows from the assumption that b is an increasing function.Then applying the Lipschitz property of b′, the a priori bound on the dual solution z1

and both primal and dual L2 lifting we have from (4)

|||z1− z11 |||2 ≤ΘKZ‖u−u1‖L2‖z1− z1

1‖L2

≤ΘKZ C∗C∗h2s0 |||u−u1||||||z1− z1

1 ||| (5)

from which the result follows.S. Pollock 40/56 Convergence of GOAFEM


Contraction

Theorem (Contraction of the combined quasi-error)

Let θ ∈ (0,1], and let Tj ,Vjj≥0 be the sequence of meshes and finite elementspaces produced by GOAFEM. Let γp > 0 as given by the primal contraction result.Then there exist constants γ > 0,τ > 0 and 0 < αD < 1, depending on the initial meshT0 and marking parameter θ such that

|||z− z2|||2 + γζ22(z2) + τ |||u−u2|||2 + τγpη

22(u2)

≤α2D

(|||z− z1|||2 + γζ

21(z1) + τ |||u−u1|||2 + τγpη

21(u1)

).

Most of the difficulty in obtaining this result lies in approximate problem changing ateach iteration of the algorithm. The mesh should be marked with respect to a residualwe can compute so we base our estimators on the approximate dual sequence.However showing contraction with respect to these estimators introduces inconvenienterror terms. So we show contraction with respect to the limiting dual problem,switching over to the approximate dual estimators for marking the grid. This introducesless inconvenient error terms.



Sketch of proof

Starting with a positive multiple γ of estimator reduction added to limiting-dualquasi-orthogonality combined with the previous two lemmas

|||z− z2|||2 + γζ22(z2,T2)≤ A|||z− z1|||2 + γBζ

21(z1) + D|||u−u1|||2

+(γ(1 + δ

−1)Λ1η20−1

)|||z2− z1|||2.

As in the usual contraction proof, choose γ to eliminate the last term.

The coefficients A and B are shown less than one for appropriate choice of parameters(this part get a little "technical.")

To control the primal error term with the coefficient D, add a positive multiple τ of theprimal contraction result. Choose τ to ensure D + α2τ < τ

τ >D

1−α2

and set

α2D := max

A,B,

D + α2τ

τ,α2

< 1.



Convergence

Corollary (Bounding the error in the goal function)

Let the combined quasi-error given by

Q2(uj , zj ) = |||z− zj |||2 + γζ2j (zj ) + τ |||u−uj |||2 + τγpη

2j (uj ).

Then there is a constant C > 0 and αD ∈ (0,1) with

|g(u)−g(uj )| ≤ CQ2j (uj , zj )≤ α

2jD CQ2

0(u0, z0).



Sketch of proof

By linearity and Galerkin orthogonality applied to the primal problem

g(u)−g(uj ) = a(u−uj , z− zj ) + 〈b(u)−b(uj ), z− zj〉+ 〈(b′(u)−Bj )(u−uj ), z〉.

The third term on the RHS represents the error induced by switching from the limitingto the linearized dual problem which enables switching between primal and dual forms.This term may be bounded by a priori estimates and

‖b′(u)−Bj‖=

∥∥∥∥Z 1

0b′(u)−b′ (uj + ξ(u−uj ))dξ

∥∥∥∥≤ Θ

2‖u−uj‖

yielding

〈(b′(u)−Bj )(u−uj ), z〉 ≤12

ΘKZ‖u−uj‖2L2

.

Then obtain

|g(u)−g(uj )| ≤12

(1 + (ΘKZ C∗+ BC∗)C∗h

2s0

)|||u−uj |||2

+12

(1 + BC∗C∗h2s0 )|||z− zj |||2.



Numerics!

Taking Ω = [0,1]2, we consider solving the following model problem using FETK:

F(u) :=−∆u + 3u3− f = 0,

with homogeneous Dirichlet boundary condition. Here the source function f is chosensuch that the exact solution is given by

u(x ,y) =sin(πx)sin(πy)

2(x−0.5)2 + 2(y−0.5)2 + 10−3 .

Finite element mesh

Figure: The mesh and finite element solution to the model problem after 12 GOAFEM iterations.S. Pollock 45/56 Convergence of GOAFEM


Numerics: error

The error in H1 and the quantity of interest with goal functiong = 100e−100((x−0.5)2+(y−0.5)2).

101 102 103 104 105 106100

101

102

103

104Error Reductions in H1 Norm

# of Vertices

H1 Erro

r

H1 Error for GOAFEMH1 Error for AFEMN 1/2

101 102 103 104 105 10610 4

10 3

10 2

10 1

100

101

102Error Reduction of |g(u uh)|

# of Vertices

|g(u

u h)|

|g(u uh)|

N 1/2

Figure: The reduction rate in the H1 norm of the error u−uh and the goal error |〈g,u−uh〉|.

We see a few oscillations in the error in the goal function which smooth out once themesh is sufficiently fine. We also see the GOAFEM algorithm decreases the error inthe goal function at a better rate than the primal energy norm error.



Quasi-optimality of the method (linear problem)

Theorem (Quasi-optimality)Let (u, f ,D) ∈ As and (z,g,D∗) ∈ At . Then there are constants Sθ,Mp,Md > 0 with

#Tk −#T0 ≤ Sθ

Mp|u, f ,D|1/s

As

(1 +

γp

c2

)1/2s

Q−1/sk (uk ,Tk )

+ Md |z,g,D∗|1/tAt

(1 +

γd

c2

)1/2t

Q−1/tk (zk ,Tk )

.

The approximation class As (respectively At ) based on the total error for s > 0

As :=

(v , f ,D)

∣∣ |v , f ,D|s := supN>0

(Nsσ(N;v , f ,D)) < ∞

.

whereσ(N;v , f ,D) := inf

T ∈TN

infvT ∈VT

(|||v− vT |||2 + osc 2

T (vT ,T )) 1

2 .

The proof follows the one outlined in [Cascon, Kreuzer, Nochetto, Siebert, 2008].



Key estimates to show quasi-optimality

Cardinality of the marked set: #Mk ≤ MpE−1/sk (uk ,Tk ) + Md E−1/t

k (zk ,Tk ).I By membership in the approximation class As (respectively At ), given ε > 0 there

is a conforming refinement Tp ∈ T and a vp ∈ VTpwith

#Tp−#T0 ≤ C|u, f ,D|1/ss ε

−1/s,

|||u− vp|||2 +osc 2Tp

(up,Tp)≤ ε2.

The ε here is chosen to be a multiple of the the total error

Ek = Ek (uk ,Tk ) =(|||u−uk |||2 +osc 2

k (uk ,Tk ))1/2

.I The total error satsifies Céa’s Lemma: E2

k (uk ,Tk )≤ CD infv∈vk E2k (v ,Tk ).

The mesh cardinality property of newest vertex bisection#Tk −#T0 ≤ Cf ∑

k−1j=0 #Mj .

Estimator as a lower bound on energy norm error yielding the equivalence ofquasi-error and total error. In particular

E−1/sj (uj ,Tj )≤

(1 +

γpc2

)1/2sQ−1/s

j (uj ,Tj ) and similarly for the dual.

Contraction of the quasi-error.



Quasi-optimality in context

Our result for GOAFEM:

#Tk −#T0 ≤ Sθ

Mp|u, f ,D|1/s

As

(1 +

γp

c2

)1/2s

Q−1/sk (uk ,Tk )

+ Md |z,g,D∗|1/tAt

(1 +

γd

c2

)1/2t

Q−1/tk (zk ,Tk )

.

AFEM as shown in [Cascon, Kreuzer, Nochetto, Siebert, 2008] for the primal problem

#Tk −#T0 ≤ Sθ

Mp|u, f ,D|1/s

As

(1 +

γp

c2

)1/2s

Q−1/sk (uk ,Tk )

.

GOAFEM as shown in [Mommer, Stevenson, 2009] for scaled Laplacian withpiecewise polynomial data

#Tk −#T0 ≤ Cε−1/(s+t) (|u|As |z|A t )1/(s+t) , |||u−uk ||||||z− zk |||< ε.










Current and future results

Our current results1 [Holst, Pollock, 2011] Convergence plus complexity analysis of a

goal-oriented adaptive method for nonsymmetric elliptic problems.2 [Holst, Pollock, Zhu, 2012] Convergence of a goal-oriented adaptive method

for semilinear problems.

We plan to extend these results to1 A coupled system of semilinear equations

1 A two-grid method where the primal problem is solved on a coarser mesh than thedual.

2 Assuming an inexact solution to the primal (nonlinear) problem.

2 A Petrov-Galerkin formulation



thank you!

thank you!



References

Bank, R., Holst, M., Szypowski, R., and Zhu, Y.Finite element error estimates for critical growth semilinear problems withoutangle conditions, 2011.

Bank, R., Holst, M., Szypowski, R., and Zhu, Y.Convergence of AFEM for semilinear problems with inexact solvers, 2011.

Holst, M., and Pollock, S.Convergence of goal oriented methods for nonsymmetric problems, 2011.

Holst, M., and Pollock, S., and Zhu, Y.Convergence of goal oriented methods for semilinear problems, 2012.

Holst, M., Tsogtgerel, G., and Zhu, Y.Local and Global Convergence of Adaptive Methods for Nonlinear PartialDifferential Equations, 2009.

Prudhomme, S., and Oden, J. T.On goal-oriented error estimation for elliptic problems: application to the control ofpointwise errors, Computer Methods in Applied Mechanics andEngineering,176,1-4,1999, 313–331.



More references

Grätsch, T., and Bathe, K.Influence functions and goal-oriented error estimation for finite element analysisof shell structures, International Journal for Numerical Methods inEngineering,63,2005, 709–736.

Estep, D., Holst, M, and Larson, M.Generalized Green’s Functions and the Effective Domain of Influence, SIAM J.Sci. Comput, 26, 2002,1314–1339.

Cascon, J.M., Kreuzer, C., Nochetto, R.H., and Siebert, K.G.Quasi-optimal convergence rate for an adaptive finite element method, SIAM J.Numer. Anal., 5, 46, 2008, 2524–2550.

Mommer, M. S., and Stevenson, R.A Goal-Oriented Adaptive Finite Element Method with Convergence Rates, SIAMJ. Numer. Anal., 47, 2, 2009, 861–886

Morin, P., Nochetto, R., and Siebert, K.,Data Oscillation and Convergence of Adaptive FEM, SIAM J. Numer. Anal., 38, 2,2000, 466–488.



Additional references

Binev, P., Dahmen, W., and DeVore, R.Adaptive Finite Element Methods with convergence rates, Numer. Math., 97, 2,2004, 219–268.

Giles, M., and Süli, E.,Adjoint methods for PDEs: a posteriori error analysis and postprocessing byduality, Acta Numerica, 2002, 145-236.

Mekchay, K., and Nochetto, R.Convergence of Adaptive Finite Element Methods for General Second OrderLinear Elliptic PDE, SINUM, 5, 43, 2005, 1803–1827.

Carey, V., Estep, D., and Tavener, S.,A posteriori analysis and adaptive error control for multiscale operatordecomposition solution of elliptic systems I: Triangular systems, SIAM J. Numer.Anal,1,47, 2009,740–761.

Ciarlet, P. G.,Finite Element Method for Elliptic Problems,2002,Society for Industrial andApplied Mathematics,Philadelphia, PA, USA



More additional references

Moon, K-S., Schwerin, E.V., Szepessy, A., and Tempone, R.Convergence rates for an adaptive dual weighted residual finite elementalgorithm, BIT, 46(2),2006,367–407.

Dahmen, W., Kunoth, A., and Vorloeper, JConvergence of adaptive wavelet methods for goal-oriented error estimationIGPM report, RWTH Aachen, 2006.



Convergence of goal-oriented adaptive finite element ... · Thesis defense Sara Pollock Thesis advisor: Michael Holst Collaborators: Michael Holst, Yunrong Zhu UCSD Mathematics May

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