basic concepts of adaptive finite element methods for

Department of Mathematics, University of HoustonInstitut fur Mathematik, Universitat Augsburg lsrmnROMUNHS0

Basic Concepts of Adaptive Finite Element

Methods for Elliptic Boundary Value Problems

Ronald H.W. Hoppe1,2

1 Department of Mathematics, University of Houston2 Institute of Mathematics, University of Augsburg

Summer School ’Challenges in Applied Control and Optimal Control’

Basque Center for Applied Mathematics

Derio, July 4-8, 2011


The Loop in Adaptive Finite Element Methods (AFEM)

Adaptive Finite Element Methods (AFEM) consist of successive loops of the cycle

SOLVE =⇒ ESTIMATE =⇒ MARK =⇒ REFINE

SOLVE: Numerical solution of the FE discretized problem

ESTIMATE: Residual and hierarchical a posteriori error estimatorsError estimators based on local averagingGoal oriented weighted dual approachFunctional type a posteriori error bounds

MARK: Strategies based on the max. error or the averaged errorBulk criterion for AFEMs

REFINE: Bisection or ’red/green’ refinement or combinations thereof


A Posteriori Error Estimation I

For a closed subspace V ⊂ H1(Ω) we assume

a(·, ·) : V × V → R

to be a bounded, V-elliptic bilinear form, i.e.,

|a(v,w)| ≤ C‖v‖k,Ω‖w‖k,Ω, v,w ∈ V, a(v,v) ≥ γ‖v‖2k,Ω, v ∈ V,

for some constants C > 0 and γ > 0. We further assume ℓ ∈ V∗ where V∗ denotes

the algebraic and topological dual of V and consider the variational equation:

Find u ∈ V such that

a(u,v) = ℓ(v) , v ∈ V.

It is well-known by the Lax-Milgram Lemma that under the above assumptions

the variational problem admits a unique solution.


A Posteriori Error Estimation II

Example. The standard example is Poisson’s equation: Let Γ = ΓD ∪ ΓN,ΓD ∩ ΓN = ∅

and V := v ∈ H1(Ω) | v|ΓD

= 0 and

a(v,w) := (∇v,∇w)0,Ω, v,w ∈ V, ℓ(v) := (f ,v)0,Ω + (g,v)0,ΓN, v ∈ V,

where f ∈ L2(Ω) and g ∈ L2

(ΓN). Then, the variational equation represents the

weak form of Poisson’s problem−∆u = f in Ω,

u = 0 on ΓD,

n · ∇u = g on ΓN,

where n stands for the unit outward normal on ΓN.


A Posteriori Error Estimation III

Example cont’d. We remark that in case ΓN = ∅ we have V = H10(Ω) (pure homoge-

neous Dirichlet boundary conditions), whereas in case ΓD = ∅ the appropriate

function space is V = v ∈ H1(Ω) | (v,χΩ)0,Ω = 0, where χΩ stands for the character-

istic function of Ω. In these cases, the V-ellipticity of the bilinear form a(·, ·)

follows from the Poincare-Friedrichs inequalities

‖v‖0,Ω ≤ C(

|v|1,Ω + |

∫

Γ

vds|)

,

‖v‖0,Ω ≤ C(

|v|1,Ω + |

∫

Ω

vdx|)

.


A Posteriori Error Estimation IV

An important issue in the theory of partial differential equations is the regularity

of a solution. For instance, considering Poisson’s problem, it is well-known that for

a convex domain Ω ⊂ R2 with piecewise smooth boundary Γ and f ∈ L2

(Ω) the

weak solution u ∈ V satisfies u ∈ V ∩ H2(Ω) and

‖u‖2,Ω ≤ C‖f‖0,Ω.

However, there is less regularity, if the domain Ω is no longer convex. A classi-

cal example is the so-called L-shaped domain: Consider the Poisson equation in

Ω := (−1, +1) × (0, 1) ∪ (−1,0) × (−1,0],

ΓD := 0 × [−1, 0] ∪ [0,1] × 0 , ΓN := Γ \ ΓD,

and assume f ≡ 0 and g such that u(r, ϕ) = r2/3sin(23ϕ) is the exact solution of

the problem. The solution belongs to V∩H5/3−ε(Ω) for any ε > 0 (but not to H5/3

(Ω)!)

and has a singularity in the origin.


A Posteriori Error Estimation VFinite element approximations are based on the Ritz-Galerkin approach: Given

a finite dimensional subspace Vh ⊂ V of test/trial functions, find uh ∈ Vh such that

a(uh,vh) = ℓ(vh), vh ∈ Vh.

Since Vh ⊂ V, the existence and uniqueness of a discrete solution uh ∈ Vh follows

readily from the Lax-Milgram Lemma. Moreover, we deduce that the error

eu := u − uh satisfies the Galerkin orthogonalitya(u − uh,vh) = 0, vh ∈ Vh,

i.e., the approximate solution uh ∈ Vh is the projection of the solution u ∈ V

onto Vh with respect to the inner product a(·, ·) on V (elliptic projection). Using

the Galerkin orthogonality, it is easy to derive the a priori error estimate

‖u − uh‖1,Ω ≤ M infvh∈Vh

‖u − vh‖1,Ω,

where M := C/γ. This result tells us that the error is of the same order as the

best approximation of the solution u ∈ V by functions from the finite dimen-

sional subspace Vh. It is known as Cea’s Lemma.


A Posteriori Error Estimation VI

The Ritz-Galerkin method also gives rise to an a posteriori error estimate in terms

of the residual r : V → R

r(v) := ℓ(v) − a(uh,v), v ∈ V.

In fact, it follows that for any v ∈ V

γ‖u − uh‖21,Ω ≤ a(u − uh,u − uh) = r(u − uh) ≤ ‖r‖−1,Ω ‖u − uh‖1,Ω,

whence

‖u − uh‖1,Ω ≤1

γsupv∈V

|r(v)|

‖v‖1,Ω

.


Reliability and Efficiency


Reliability and Efficiency of Error Estimators I

Definition. An error estimator ηh is called reliable, if it provides an upper bound

for the error up to data oscillations oscrelh , i.e., if there exists a constant Crel > 0,

independent of the mesh size h of the underlying triangulation, such that

‖eu‖a ≤ Crel ηh + oscrelh .

On the other hand, an estimator ηh is said to be efficient, if up to data oscilla-

tions osceffh it gives rise to a lower bound for the error, i.e., if there exists a constant

Ceff > 0, independent of the mesh size h of the underlying triangulation, such that

ηh ≤ Ceff ‖eu‖a + osceffh .

Finally, an estimator ηh is called asymptotically exact, if it is both reliable and effi-

cient with Crel = C−1eff .


Reliability and Efficiency of Error Estimators IIRemark. The notion ’reliability’ is motivated by the use of the error estimator in

error control. Given a tolerance tol, an idealized termination criterion would be

‖eu‖a ≤ tol.

Since the error ‖eu‖a is unknown, we replace it with the upper bound, i.e.,

Crel ηh + oscrelh ≤ tol.

We note that the termination criterion both requires the knowledge of Crel and

the incorporation of the data oscillation term oscrelh . In the special case Crel = 1

and oscrelh ≡ 0, it reduces to

ηh ≤ tol.

An alternative, but less used termination criterion is based on the lower bound,

i.e., we require 1

Ceff

(

ηh − osceffh

)

≤ tol.

Typically, this criterion leads to less refinement and thus requires less computa-

tional time which motivates to call the estimator efficient.


Residual-Type A Posteriori Error Estimation


The Role of the ResidualThe error estimate

‖u − uh‖1,Ω ≤1

γsupv∈V

|r(v)|

‖v‖1,Ω

shows that in order to assess the error ‖eu‖a we are supposed to evaluate the norm

of the residual with respect to the dual space V∗, i.e.,

‖r‖V∗ := supv∈V\0

|r(v)|

‖v‖a

.

In particular, we have the equality

‖r‖V∗ = ‖eu‖a,

whereas for the relative error of r(v),v ∈ V, as an approximation of ‖eu‖a we obtain

(‖eu‖a − r(v))

‖eu‖a

=1

2‖v −

eu

‖eu‖a

‖2a, v ∈ V with ‖v‖a = 1.

The goal is to obtain lower and upper bounds for ‖r‖V∗ at relatively low compu-

tational expense.


Model problem: Let Ω be a bounded simply-connected polygonal domain in Eucli-

dean space lR2 with boundary Γ = ΓD ∪ ΓN , ΓD ∩ ΓN = ∅ and consider the ellip-

tic boundary value problem

Lu := − ∇ · (a ∇ u) = f in Ω ,

u = 0 on ΓD , n · a ∇u = g on ΓN,

where f ∈ L2(Ω) , g ∈ L2(ΓN) and a = (aij)2i,j=1 is supposed to be a matrix-valued func-

tion with entries aij ∈ L∞(Ω), that is symmetric and uniformly positive definite. The

vector n denotes the exterior unit normal vector on ΓN. Setting

H10,ΓD

(Ω) := v ∈ H1(Ω) | v |ΓD= 0 ,

the weak formulation is as follows: Find u ∈ H10,ΓD

(Ω) such that

a(u,v) = ℓ(v) , v ∈ H10,ΓD

(Ω),where

a(v,w) :=

∫

Ω

a ∇v · ∇w dx, ℓ(v) :=

∫

Ω

f v dx +

∫

ΓN

g v dσ , v ∈ H10,ΓD

(Ω).


FE Approximation: Given a geometrically conforming simplicial triangulation Th of

Ω, we denote by

S1,ΓD(Ω;Th) := vh ∈ H1

0,ΓD(Ω) | vh |T∈ P1(K) , T ∈ Th

the trial space of continuous, piecewise linear finite elements with respect to Th.

Note that Pk(T) , k ≥ 0, denotes the linear space of polynomials of degree ≤ k on T.

In the sequel we will refer to Nh(D) and Eh(D) , D ⊆ Ω as the sets of vertices and

edges of Th on D. We further denote by |T| the area, by hT the diameter of an

element T ∈ Th, and by hE = |E| the length of an edge E ∈ Eh(Ω ∪ ΓN). We

refer to fT := |T|−1∫

Tfdx the integral mean of f with respect to an element T ∈ Th

and to gE := |E|−1∫

Egds the mean of g with respect to the edge E ∈ Eh(ΓN).

The conforming P1 approximation reads as follows: Find uh ∈ S1,ΓD(Ω; Th) such that

a(uh,vh) = ℓ(vh), vh ∈ S1,ΓD(Ω; Th).


Representation of the Residual I

The residual r is given by

r(v) :=

∫

Ω

f v dx +

∫

ΓN

g vds − a(uh,v) , v ∈ V.

Applying Green’s formula elementwise yields

a(uh,v) =

∑

T∈Th

∫

T

a ∇uh · ∇v dx =

∑

E∈Eh(Ω)

∫

E

[n · a ∇uh] v ds +

∑

E∈Eh(ΓN)

∫

E

n · a ∇uh v ds,

where [n · a ∇uh] denotes the jump of the normal derivative of uh across E ∈ Eh(Ω)

and where we have used that ∆uh ≡ 0 on T ∈ Th, since uh|T ∈ P1(T). We thus obtain

r(v) :=

∑

T∈Th

rT(v) +

∑

E∈Eh(Ω∪ΓN)

rE(v).


Representation of the Residual II

Here, the local residuals rT(v),T ∈ Th, are given by

rT(v) :=

∫

T

(f − Luh)v dx,

whereas for rE(v) we have

rE(v) := −

∫

E

[n · a ∇uh]v ds, E ∈ Eh(Ω),

rE(v) :=

∫

E

(

g − n · a ∇uh

)

v ds, E ∈ Eh(ΓN).


A Posteriori Error Estimator and Data Oscillations

The error estimator ηh consists of element residuals ηT,T ∈ Th, and edge resi-

duals ηE,E ∈ EH(Ω ∪ ΓN), according to

ηh :=

(

∑

T∈Th

η

2T +

∑

E∈EH(Ω∪ΓN)

η

2E

)1/2

,

where ηT and ηE are given by

ηT := hT ‖fT − Luh‖0,T , T ∈ Th,

ηE :=

h1/2

E ‖[n · a ∇uh]‖0,E , E ∈ Eh(Ω),

h1/2

E ‖gE − n · a ∇uh‖0,E , E ∈ Eh(ΓN)

.

The a posteriori error analysis further invokes the data oscillations

osch :=

(

∑

T∈Th

osc2T(f) +

∑

E∈Eh(ΓN)

osc2E(g)

)1/2

,

where oscT(f) and oscE(g) are given by

oscT(f) := hT ‖f − fT‖0,T, oscE(g) := h1/2

E ‖g − gE‖0,E.

Department of Mathematics, University of HoustonInstitut fur Mathematik, Universitat Augsburg lsrmnROMUNHS0Clement’s Quasi-Interpolation Operator I

For p ∈ Nh(Ω)∪Nh(ΓN) we denote by ϕp the basis function in S1,ΓD(Ω;Th) with suppor-

ting point p, and we refer to Dp as the set

Dp :=

⋃

T ∈ Th | p ∈ Nh(T) .

We refer to πp as the L2-projection onto P1(Dp), i.e.,

(πp(v),w)0,Dp = (v,w)0,Dp , w ∈ P1(Dp),

where (·, ·)0,Dp stands for the L2-inner product on L2(Dp) × L2

(Dp). Then,

Clement’s interpolation operator PC is defined as follows

PC : L2(Ω) −→ S1,ΓD

(Ω, Th), PCv :=

∑

p∈Nh(Ω)∪Nh(ΓN)

πP(v) ϕP.

Department of Mathematics, University of HoustonInstitut fur Mathematik, Universitat Augsburg lsrmnROMUNHS0Clement’s Quasi-Interpolation Operator II

Theorem. Let v ∈ H10,ΓD

(Ω). Then, for Clement’s interpolation operator there holds

‖PC v‖0,T ≤ C ‖v‖0,D

(1)

T

, ‖PC v‖0,E ≤ C ‖v‖0,D

(1)

E

, ‖∇PCv‖0,T ≤ C ‖∇v‖0,D

(1)

T

,

‖v − PC v‖0,T ≤ C hT ‖v‖1,D

(1)

T

, ‖v − PC v‖0,E ≤ C h1/2

E ‖v‖1,D

(1)

E

.

Further, we have(

∑

T∈Th

‖v‖2

µ,D(1)

K

)1/2

≤ C ‖v‖µ,Ω, 0 ≤ µ ≤ 1,

(

∑

E∈Eh(Ω)∪Eh(ΓN)

‖v‖2

µ,D(1)

E

)1/2

≤ C ‖v‖µ,Ω, 0 ≤ µ ≤ 1.

where D(1)

T :=⋃

T′ ∈ Th | Nh(T′)∩Nh(T) 6= ∅ , D

(1)

E :=⋃

T′ ∈ Th | Nh(E)∩Nh(T′) 6= ∅ .


Reliability of the A posteriori Error Estimator I

Theorem. There exist constants ΓR and Γosc > 0 depending only on the shape regula-

rity of Th such that

‖eu‖a ≤ ΓR ηh + Γosc osch.

Proof. Setting v = eu, we have

‖eu‖a = a(eu, eu) = r(eu) = r(PCeu) + r(eu − PCeu).

For the first term on the right-hand side, by Galerkin orthogonality we obtain

r(PCeu) =

∫

Ω

f PCeu dx +

∫

ΓN

g PCeu ds − a(uh,PCeu) = 0.


Proof cont’d. On the other hand, for the second term on the right-hand side of

Green’s formula yields

r(eu − PCeu) =

∫

Ω

f (eu − PCeu) dx +

∫

ΓN

g (eu − PCeu) ds +

∑

T∈Th

∫

T

∇ · a ∇ uh︸︷︷︸

= −Luh

(eu − PCeu) dx

−∑

T∈Th

∫

∂T

n∂T · a ∇uh (eu − PCeu) ds =

∑

T∈Th

∫

T

(fT − Luh) (eu − PCeu) dx

+

∑

E∈Eh(Ω)

∫

E

[nE · a ∇uh] (eu − PCeu) ds +

∑

E∈Eh(ΓN)

∫

E

(gE − nE · a ∇uh) (eu − PCeu) ds

+

∑

T∈Th

∫

T

(f − fT) (eu − PCeu) dx +

∑

E∈Eh(ΓN)

∫

E

(g − gE) (eu − PCeu) ds .


Proof cont’d. In view of the local approximation properties of Clement’s quasi-

interpolation operator, it follows that

r(eu − PCeu) ≤ C(

∑

T∈Th

h2T ‖fT − Luh‖

20,T)

1/2 ‖eu‖1,Ω

+

∑

E∈Eh(Ω)

hE ‖[nE · a ∇uh]J‖20,E)

1/2 ‖eu‖1,Ω

+

∑

E∈Eh(ΓN)

hE ‖gE − nE · a ∇uh‖20,E)

1/2 ‖eu‖1,Ω + (

∑

T∈Th

h2T ‖f − fT‖

20,T)

1/2 ‖eu‖1,Ω

+

∑

E∈Eh(ΓN)

hE ‖g − gE‖20,E)

1/2 ‖eu‖1,Ω

)

,

from which we may conclude.


Element and Edge Bubble Functions I

The element bubble function ψT is defined by means of the barycentric coordi-

nates λTi , 1 ≤ i ≤ 3, according to

ψT := 27 λT1 λ

T2 λ

T3 .

Note that supp ψT = Tint, i.e., ψT |∂T= 0 , T ∈ Th. On the other hand, for E ∈ Eh(Ω)∪

Eh(ΓN) and T ∈ Th such that E ⊂ ∂T and pEi ∈ Nh(E) , 1 ≤ i ≤ 2, we introduce

the edge-bubble functions ψE

ψE := 4 λT1 λ

T2 .

Note that ψE |E′= 0 for E′ ∈ Eh(T),E′ 6= E.

Department of Mathematics, University of HoustonInstitut fur Mathematik, Universitat Augsburg lsrmnROMUNHS0Element and Edge Bubble Functions II

The bubble functions ψT and ψE have the following important properties that can

be easily verified taking advantage of the affine equivalence of the finite elements:

Lemma. There holds

‖ph‖20,T ≤ C

∫

T

p2h ψT dx, ph ∈ P1(T),

‖ph‖20,E ≤ C

∫

E

p2h ψE dσ, ph ∈ P1(E),

| ph ψT |1,T ≤ C h−1T ‖ph‖0,T, ph ∈ P1(T),

‖ph ψT‖0,T ≤ C ‖ph‖0,T, ph ∈ P1(T),

‖ph ψE‖0,E ≤ C ‖ph‖0,E , ph ∈ P1(E).

Department of Mathematics, University of HoustonInstitut fur Mathematik, Universitat Augsburg lsrmnROMUNHS0Element and Edge Bubble Functions III

For functions ph ∈ P1(E) , E ∈ Eh(Ω) ∪ Eh(ΓN) we further need an extension pEh ∈ L2

(T)

where T ∈ Th such that E ⊂ ∂T. For this purpose we fix some E′ ⊂ ∂T , E′ 6= E, and

for x ∈ T denote by xE that point on E such that (x − xE) ‖ E′. For ph ∈ P1(E) we

then set

pEh := ph(xE).

Further, for E ∈ Eh(Ω) ∪ Eh(ΓN) we define D(2)

E as the union of elements T ∈ Th con-

taining E as a common edge

D(2)

E :=

⋃

K ∈ Th | E ∈ Eh(T) .

Department of Mathematics, University of HoustonInstitut fur Mathematik, Universitat Augsburg lsrmnROMUNHS0Element and Edge Bubble Functions IV

Lemma. There holds

| pEh ψE |

1,D(2)

E

≤ C h−1/2

E ‖ph‖0,e, ph ∈ P1(E),

‖pEh ψE‖0,D

(2)

E

≤ C h1/2

E ‖ph‖0,E, ph ∈ P1(E).

Further, for all v ∈ V and µ = 0,1 there holds

(

∑

E∈Eh(Ω)∪Eh(ΓN)

h1−µE ‖v‖2

µ,D(2)

E

)1/2 ≤ C (

∑

T∈Th

h1−µT ‖v‖2

µ,T)1/2

.


Efficiency of the A posteriori Error Estimator I

Theorem. There exist constants γR,γE > 0, depending only on the shape regularity

of Th such that

γR ηh − γE osch ≤ ‖e‖1,Ω.

Proof. The proof will be given by a series of lemmas establishing the local efficiency

of the estimator.


Lemma. Let T ∈ Th. Then there holds:

hT ‖fT − Luh‖0,T ≤ C ‖e‖1,T + hT ‖f − fT‖0,T.

Proof. We set ph := fT. Observing ψT|∂T = 0, by Green’s formula

a|T(uh,ph ψT) = −

∫

T

∇ · (a ∇uh) ph ψT dx +

∫

∂T

n∂T · a ∇uh ph ψT︸︷︷︸

= 0

ds.

Denoting by πhψT the L2-projection onto the linear space of elementwise constants

and taking advantage of the properties of the bubble functions, it follows that

‖fT − Luh‖20,T ≤ C

∫

T

(fT − Luh) πh ψT dx =

(

∫

T

f πh ψT dx − a|T(uh,πh ψT)

+

∫

T

(πhf − f) πh ψT dx)

=

(

a|T(eu,πh ψT) +

∫

T

(fT − f) πh ψT dx)

≤ C h−1T ‖eu‖1,T ‖ph‖0,T + ‖fT − f‖0,T ‖ph‖0,T,


Lemma. Let E ∈ Eh(Ω). Then there holds:

h1/2

E ‖[nE · a ∇uh]‖0,E ≤ C ‖e‖1,D

(2)

E

+ hE ‖f − fT‖0,D

(2)

E

+ hE ‖fT − Luh‖0,D

(2)

E

.

Proof. We set pEh := [nE · a ∇uh]. In view of ψE|E′ = 0,E′ 6= E,, Green’s formula gives

∫

∂D(2)

E

n∂D

(2)

E

· a ∇uh pEh ψE ds = a|

D(2)

E

(uh,pEh ψE) +

∫

D(2)

E

∇ · a ∇uh︸︷︷︸

= −Luh

pEh ψE dx,

where pEhψE is the L2-projection onto the edgewise constants.


Proof cont’d. If we use the properties of the bubble functions, it follows that

‖[n∂D

(2)

E

· a ∇uh]‖20,E ≤ C

∫

E

[n∂D

(2)

E

· a ∇uh] pEh ψE ds =

∫

∂D(2)

E

[n∂D

(2)

E

· a ∇uh] pEh ψE ds

=

(

a|D

(2)

E

(uh,pEh ψE) −

∫

D(2)

E

f pEh ψEdx +

∫

D(2)

E

(f − fT) pEh ψEdx +

∫

D(2)

E

(fT − Luh) pEh ψEdx

)

= −a|D

(2)

E

(e,pEh ψE) +

(

∫

D(2)

E

(f − fT) pEh ψEdx +

∫

D(2)

E

(fT − Luh) pEh ψEdx

)

≤ C h−1/2

E ‖eu‖1,D

(2)

E

‖pEh‖0,E + h

1/2

E ‖f − fT‖0,D

(2)

E

‖pEh‖0,E + h

1/2

E ‖fT − Luh‖0,D

(2)

E

‖pEh‖0,E.

from which the assertion can be easily deduced.


Lemma. Let E ∈ Eh(ΓN). Then there holds:

h1/2

E ‖gE − nE · ∇uh‖0,E ≤

C ‖e‖1,D

(2)

E

+ h1/2

E ‖g − gE‖0,E + hE ‖f − fT‖0,D

(2)

E

+ hE‖fT − Luh‖0,D

(2)

E

.

Proof. We set pEh := gE − nE · a ∇uh. Observing ψE|E′ = 0,E′ 6= E, by Green’s formula

we obtain∫

E

nE · a ∇uh pEh ψE ds =

∫

∂D(2)

E

n∂D

(2)

E

· a ∇uh pEh ψE ds

= a|D

(2)

E

(uh,pEh ψE) +

∫

D(2)

E

∇ · a ∇uh︸︷︷︸

= −Luh

pEh ψE dx.


Goal-Oriented Dual Weighted Approach

Department of Mathematics, University of HoustonInstitut fur Mathematik, Universitat Augsburg lsrmnROMUNHS0Goal-Oriented Dual Weighted Approach I

The goal oriented dual weighted approach allows to control the error eu := u − uh

with respect to a rather general error functional or output functional

J : V ⊆ H1(Ω) → lR.

The goal oriented dual weighted approach strongly uses the solution z ∈ V of the

associated dual problema(v, z) = J(v) , v ∈ V,

and its finite element approximation zh ∈ Vh, i.e.,

a(vh, zh) = J(vh), vh ∈ Vh.

Using Galerkin orthogonality, we readily deduce that

J(eu) = a(eu, z) = a(eu, z − vh) = r(z − vh), vh ∈ Vh,

where r(·) stands for the residual with respect to the computed finite element

approximation uh.


Goal-Oriented Dual Weighted Approach IITheorem. Let uh ∈ Vh := S1,Γ(Ω; Th(Ω)) be the conforming P1 approximation of

the solution u ∈ H10(Ω) of Poisson’s equation with f ∈ L2

(Ω) and homogeneous

Dirichlet boundary data. Then, the following error representation holds true

J(eu) =

∑

T∈Th(Ω)

(

(rT, z − vh)0,T + (r∂T, z − vh)0,∂T

)

, vh ∈ Vh,

where the element residuals rT and the edges residuals r∂T are given by

rT := f , T ∈ Th(Ω), r∂T|E :=

12

νE · [∇uh] , E ∈ Eh(∂T ∩ Ω)

0 , E ∈ Eh(∂T ∩ Γ)

Moreover, we have the error estimate

|J(eu)| ≤ ηDW :=

∑

T∈Th(Ω)

ωT ρT,

where for vh ∈ Vh the element residuals ρT and the weights ωT read

ρT :=

(

‖rT‖20,T + h−1

T ‖r∂T‖

20,∂T

)1/2

, ωT :=

(

‖z − vh‖20,T + hT ‖z − vh‖

20,∂T

)1/2

.


Goal-Oriented Dual Weighted Approach III

We remark that the previous result is not really a posteriori, since the solu-

tion z ∈ V of the dual solution is not known. Therefore, information about

the weights ωT,T ∈ Th(Ω) has to be provided either by means of an a priori

analysis or by the numerical solution of the dual problem.

Theorem. Under the assumptions of the previous theorem let the error func-

tional be given by

J(v) :=(∇v, ∇eu)0,Ω

‖∇eu‖0,Ω

, v ∈ V.

Then, there holds‖∇eu‖0,Ω ≤ C

(

∑

T∈Th(Ω)

h2T ρ

2T

)1/2

.


Proof. The dual solution z ∈ V satisfies

a(v, z) =(∇v,∇eu)0,Ω

‖∇eu‖0,Ω

, v ∈ V,

from which we readily deduce the a priori bound

‖∇z‖0,Ω ≤ 1.

In view of the basic error estimate it follows that

J(eu) = ‖∇eu‖0,Ω ≤(

∑

T∈Th(Ω)

h2T ρ

2T

)1/2 (

∑

T∈Th(Ω)

h−2T ω

2T

)1/2

.

Choosing vh = PCz, where PC is Clement’s quasi-interpolation operator, we find

infvh∈Vh

(

∑

T∈Th(Ω)

(h−2T ‖z − vh‖

20,T + h−1

T ‖z − vh‖20,∂T

)1/2

≤ C ‖∇z‖0,Ω.

Using the last inequality in the previous one and observing the error represen-

tation gives the assertion.


Goal-Oriented Dual Weighted Approach IV

Theorem. Consider the conforming P1 approximation of Poisson’s equation

under homogeneous Dirichlet boundary conditions and assume that the solution

u ∈ V := H10(Ω) is 2-regular. Using the the error functional

J(v) :=(v, eu)0,Ω

‖eu‖0,Ω

, v ∈ V,

gives rise to the a posteriori error estimate

‖eu‖0,Ω ≤ C(

∑

T∈Th(Ω)

h4T ρ

2T

)1/2

.


Goal-Oriented Dual Weighted Approach V

Finally, we apply the goal-oriented dual weighted approach to the pointwise

estimation of the error at some point a ∈ Ω. Given some tolerance ε > 0,

we consider the ballK

ε(a) := x ∈ Ω | |x − a| < ε

around the point a and define the regularized error functional

J(v) := |Kε(a)|−1

∫

Kε(a)

v dx.

The dual solution z of a(v, z) = J(v) behaves like a regularized Green’s function

z(x) ∼ log(r(x)) , r(x) :=

√

|x − a|2 + ε2.

With the residual ρT we obtain

|(u − uh)(a)| ∼∑

T∈Th(Ω)

h3T

r2T

ρT, rT := maxx∈T r(x).

basic concepts of adaptive finite element methods for

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