adaptive fem

8/12/2019 Adaptive Fem

1/144

Adaptive Finite Element MethodsLecture Notes Winter Term 2011/12

R. Verfürth

Fakultät für Mathematik, Ruhr-Universität Bochum


2/144


3/144

Contents

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

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

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

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

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

II.1.9. Residual a posteriori error estimate 383


4/144

4 CONTENTS

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

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

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

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

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

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

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

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

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

III.2.3. Grid hierarchy 110


5/144

CONTENTS 5

III.3. Numerical examples 110

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

IV.4.4. Restriction 136

Bibliography 139

Index 141


6/144


7/144

CHAPTER I

Introduction

I.1. Motivation

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

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

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

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

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

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

Algorithm I.1.1. (General adaptive algorithm)

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

7


8/144

8 I. INTRODUCTION

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

T 0 representing sufficiently

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

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

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

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

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

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

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

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

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

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

mate,• a refinement strategy which determines which elements have

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

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

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

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

2π, which vanishes on the straight parts ΓD of the boundary

∂ Ω, and which has normal derivative 23 sin( 2

3ϕ) on the curved part ΓN

of ∂ Ω. Using polar co-ordinates, one easily checks that

u = r2/3

sin(

2

3ϕ).


9/144

I.1. MOTIVATION 9

Figure I.1.1. Triangulation obtained by uniform refinement

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

Find a continuous piecewise linear function uT such that Ω

∇uT · ∇vT =

ΓN

2

3 sin(

2

3ϕ)vT

holds for all continuous piecewise linear functions vT .The triangulation of figure I.1.1 is obtained by five uniform refinementsof an initial triangulation T 0 which consists of three right-angled isosce-les triangles with short sides of unit length. In each refinement stepevery triangle is cut into four new ones by connecting the midpointsof its edges. Moreover, the midpoint of an edge having its two end-points on ∂ Ω is projected onto ∂ Ω. The triangulation in figure I.1.2is obtained from T 0 by applying six steps of the adaptive refinementstrategy described above using the error estimator ηR,K of section II.1.9(p. 38). A triangle K ∈ T k is divided into four new ones if

ηR,K ≥ 0.5 maxK ∈T k

ηR,K

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

ε = ∇(u − uT )

∇u .


10/144

10 I. INTRODUCTION

It clearly shows the advantages of the adaptive refinement strategy.

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

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

Figure I.1.2. Triangulation obtained by adaptive refinement

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

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

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

ALF in particular offers the following options:

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

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

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

lutions having interior and boundary layers,

http://www.rub.de/num1/files/ALF.htmlhttp://www.rub.de/num1/files/ALFUserGuide.pdfhttp://www.rub.de/num1/files/ALFUserGuide.pdfhttp://www.rub.de/num1/files/ALF.html


11/144

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

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

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

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

ment based on various a posteriori error estimators.

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

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

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

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

Here p is a continuously differentiable function with

p(x) ≥ p > 0

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

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

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

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

• Integrate the result from 0 to 1: 1

0

−( pu)(x)v(x) + q (x)u(x)v(x)dx = 10

f (x)v(x)dx.

• Use integration by parts for the term containing derivatives:

− 1

0

( pu)(x)v(x)dx

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

− p(1)u(1) v(1) =0

+

10

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

= 1

0

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


12/144

12 I. INTRODUCTION

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

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

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

0

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

=0−u(0) v(0)

=0− 1

0

u(x)v(x)dx

= − 10

u(x)v(x)dx.

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

0

w(x)v(x)dx = − 1

0

u(x)v(x)dx.

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

w(x) =

2 for 0 < x < 1

2

−2 for 12

< x


13/144

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

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

whose weak derivative exists and is contained in L2(0, 1)too.H 10 (0, 1) denotes the Sobolev space of all functions v inH 1(0, 1) which satisfy v(0) = 0 and v(1) = 0.

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

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

of 1x

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

Every continuous, piecewise continuously differentiable function is con-tained in H 1(0, 1).The function v(x) = 1−|2x−1| is contained in H 10 (0, 1) (cf. fig. I.2.1).The function v(x) = 2

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

of 1x

=

v(x))2 is not finite.

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

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

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

given by:

Find u ∈ H 10 (0, 1) such that for all v ∈ H 10 (0, 1) there holds 10

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

dx =

10

f (x)v(x)dx.

It has the following properties:

It admits a unique solution.Its solution is the unique minimum in H 10 (0, 1) of the energy


14/144

14 I. INTRODUCTION

function

1

2

10

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

dx − 1

0

f (x)u(x)dx.

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

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

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

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

Find a trial function uT ∈ S k,00 (T ) such that every test function vT ∈ S k,00 (T ) satisfies 1

0

p(x)uT (x)v

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

dx =

10

f (x)vT (x)dx.


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

T ) of the energy

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

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


15/144

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

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

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

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

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

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

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

ΓD Dirichlet part of Ω, supposed to be non-empty;ΓN Neumann part of Ω, may be empty;

n exterior unit normal to Ω;

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

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

S, T, . . . tensor-fields with values in Rd×d;

I unit tensor;

∇ gradient;div divergence;

div u =di=1

∂ui∂xi

;

div T = di=1

∂T ij∂xi

1≤ j≤d

;

∆ = div ∇ Laplace operator;D(u) =

1

2

∂ui∂x j

+ ∂ u j∂xi 1≤i,j≤d

deformation tensor;

u · v inner product;


16/144

16 I. INTRODUCTION

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

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

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

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

Γ

pu · ndS =

Ω

∇ p · udx +

Ω

p div udx, Γ

n · T · udS =

Ω

(div T) · udx +

Ω

T : D(u)dx.

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

A = {x ∈ R3 : x21 + x22 + x23


17/144


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

Dαϕ = ∂ α1+...+αdϕ

∂xα11 . . . ∂ xαdd

.

Given two functions ϕ, ψ ∈ C ∞0 (Ω), the Gauss theorem for integralsyields for every multi-index α ∈ Nn the identity

Ω

Dαϕψdx = (−1)α1+...+αd

Ω

ϕDαψ.

This identity motivates the definition of the weak derivatives:

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

Ω

ψρdx = (−1)α1+...+αd

Ω

ϕDαρ

holds for all functions ρ ∈ C ∞0 (Ω). In this case we writeψ = Dαϕ.

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

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

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

H k

(Ω) = {ϕ ∈ L2

(Ω) : Dα

ϕ ∈ L2

(Ω) for all α ∈ Nd

with α1 + . . . + αd ≤ k},

|ϕ|k =

α∈Ndα1+...+αd=k

Dαϕ2L2(Ω) 1

2

,

ϕk = k=0

|ϕ|2 1

2

=

α∈Ndα1+...+αd≤k

Dαϕ2L2(Ω) 1

2

,

H 10 (Ω) =

{ϕ

∈ H 1(Ω) : ϕ = 0 on Γ

},


18/144

18 I. INTRODUCTION

H 1D(Ω) =

{ϕ

∈ H 1(Ω) : ϕ = 0 on ΓD

},

H 12 (Γ) = {ψ ∈ L2(Γ) : ψ = ϕ

Γfor some ϕ ∈ H 1(Ω)},

ψ 12,Γ = inf {ϕ1 : ϕ ∈ H 1(Ω), ϕ

Γ

= ψ}.

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

Remark I.3.5. The space H 1

2 (Γ) is called trace space of H 1(Ω), itselements are called traces of functions in H 1(Ω).

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

general not continuous and do not admit point values (cf. ExampleI.3.7 below). A function, however, which is piecewise differentiable isin H 1(Ω) if and only if it is globally continuous. This is crucial forfinite element functions.

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

x21 + x

22 + x

23).

Example I.3.8. Consider the open unit ballΩ = {x ∈ Rd : x21 + . . . + x2d 1 − d

2 if d > 2.

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

equalities are fundamental:

ϕ0 ≤ cΩ|ϕ|1 for all ϕ ∈ H 1D(Ω),Friedrichs inequality

ϕ0 ≤ cΩ|ϕ|1 for all ϕ ∈ H 1(Ω) with

Ω

ϕ = 0

Poincaré inequality .

The constants cΩ and cΩ depend on the domain Ω and are propor-

tional to its diameter.


19/144


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

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

Any partition T has to satisfy the following conditions:

• Ω ∪ Γ is the union of all elements in T .• (Affine equivalence) Each K ∈ T is either a trian-

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

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

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

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

With every partition T we associate its shape parameter

CT = maxK ∈T

hK ρK

.

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

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

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

N K : the vertices of K ,

E K : the edges or faces of K ,


20/144

20 I. INTRODUCTION

Figure I.3.1. Mixture of triangular and quadrilateral elements

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

N T = K ∈T N K ,

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

K ∈T

E K ,

N E : the vertices of an edge or face E ∈ E T , N T ,Γ: the vertices on the boundary, N T ,ΓD: the vertices on the Dirichlet boundary, N T ,ΓN : the vertices on the Neumann boundary, N T ,Ω: the vertices in the interior of Ω,E T ,Γ: the edges or faces contained in the boundary,E T ,ΓD: the edges or faces contained in the Dirichlet

boundary,E T ,ΓN : the edges or faces contained in the Neumann

boundary,E T ,Ω: the edges or faces having at least one endpoint

in the interior of Ω.

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

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

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

ωK =E K∩E K=∅

K , ωK = N K∩N K=∅

K ,

ωE =

E ∈E KK ,

ωE =

N E∩N K=∅K ,


21/144


ωx = x∈N K K .Due to the shape-regularity of T the diameter of any of these sets

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

•

Figure I.3.2. Some domains ωK , ωK , ωE , ωE , and ωx

Figure I.3.3. Some examples of domains ωx

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

|α|1 = α1 + . . . + αd,|α|∞ = max{αi : 1 ≤ i ≤ d},

xα

= xα11 · . . . · x

αdd .


22/144

22 I. INTRODUCTION

Denote by

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

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

Rk(

K ) =

span{xα : |α|1 ≤ k} ,if K is the reference simplex,span{xα : |α|∞ ≤ k} ,if K is the reference cube

and setRk(K ) =

p ◦ F −1K : p ∈ Rk .With this notation we define finite element spaces by

S k,−1(T ) = {ϕ : Ω → R : ϕK

∈ Rk(K ) for all K ∈ T },S k,0(T ) = S k,−1(T ) ∩ C (Ω),S k,00 (T ) = S k,0(T ) ∩ H 10 (Ω) = {ϕ ∈ S k,0(T ) : ϕ = 0 on Γ}.S k,0D (

T ) = S k,0(

T )

∩H 1D(Ω) =

{ϕ

∈ S k,0(

T ) : ϕ = 0 on ΓD

}.

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

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

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

For the reference square on the other hand, we have

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

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

fined above satisfy the following approximation properties:

inf ϕT ∈S k,−1(T )

ϕ − ϕT 0 ≤ chk+1|ϕ|k+1 ϕ ∈ H k+1(Ω), k ∈ N,

inf ϕT ∈S k,0(T )

|ϕ − ϕT | j ≤ chk+1− j|ϕ|k+1 ϕ ∈ H k+1(Ω),

j

∈ {0, 1

}, k

∈ N∗,


23/144


inf ϕT ∈S

k,00 (T ) |

ϕ

−ϕT | j

≤ chk+1− j

|ϕ

|k+1 ϕ

∈ H k+1(Ω)

∩H 10 (Ω),

j ∈ {0, 1}, k ∈ N∗.

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

For any vertex x ∈ N T the associated nodal shape function is de-noted by λx. It is the unique function in S

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

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

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

from the co-ordinates of the element’s vertices.

a0 a0a1 a1

a2 a2a3

Figure I.3.4. Enumeration of vertices of triangles andparallelograms

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

a0, . . . , λ

a2 are given by

λai

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

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

a0, . . . , λ

a3 are given by

λai

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

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

i = 0, . . . , 3,

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

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


24/144

24 I. INTRODUCTION

λa0

, . . . , λa3

are given by

λai(x) = det(x

−ai+1 , ai+2

−ai+1 , ai+3

−ai+1)

det(ai − ai+1 , ai+2 − ai+1 , ai+3 − ai+1) i = 0, . . . , 3,where all indices have to be taken modulo 4.(4) Consider a parallelepiped K with vertices a0, . . . , a7 enumerated asin Figure I.3.5. Then the restrictions to K of the nodal shape functionsλa0

, . . . , λa7

are given by

λai

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

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

−ai+2 , ai+3

−ai+2 , ai+6

−ai+2)

·

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

where all indices have to be taken modulo 8.

a0 a0a1 a1

a3

a2 a3

a7

a4

a6

a5

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

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

The functions λx, x ∈ N T , form a bases of S 1,0(T ). The basesof higher-order spaces S k,0(T ), k ≥ 2, consist of suitable products of functions λx corresponding to appropriate vertices x.

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

numbered as in example I.3.12 (1). Then the nodal basis of S 2,0(T )K

consists of the functions

λai [λai − λai+1 − λai+2] i = 0, . . . , 2


25/144


4λai

λai+1

i = 0, . . . , 2,

where the functions λa are as in example I.3.12 (1) and where allindices have to be taken modulo 3. An other basis of S 2,0(T )

K , called

hierarchical basis , consists of the functions

λai

i = 0, . . . , 2

4λai

λai+1

i = 0, . . . , 2.

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

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

consists of

the functions

λai [λai − λai+1 + λai+2 − λai+3] i = 0, . . . , 34λ

ai[λ

ai+1 − λ

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

16λa0

λa2

where the functions λa

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

indices have to be taken modulo 3. The hierarchical basis of S 2,0(T )K

consists of the functions

λai

i = 0, . . . , 3

4λai

[λai+1

− λai+2

] i = 0, . . . , 3

16λa0λa2 .(3) Consider a again a tetrahedron K with its vertices numbered as in

example I.3.12 (3). Then the nodal basis of S 2,0(T )K

consists of the

functions

λai

[λai − λ

ai+1 − λ

ai+2 − λ

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

4λai

λaj

0 ≤ i < j ≤ 3,where the functions λ

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

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

λai

i = 0, . . . , 3

4λai

λaj

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

the quasi-interpolation operator I T : L1(Ω) → S 1,0D (T ) which is definedby

I T ϕ =

x∈N T ,Ω∪N T ,ΓN

λx1

|ωx| ωx

ϕdx.


26/144

26 I. INTRODUCTION

Here, |ωx| denotes the area, if d = 2, respectively volume, if d = 3, of the set ωx.

The operator I T satisfies the following local error estimates for all ϕ ∈H 1D(Ω) and all elements K ∈ T :

ϕ − I T ϕL2(K ) ≤ cA1hK ϕH 1(ωK),ϕ − I T ϕL2(∂K ) ≤ cA2h

1

2

K ϕH 1(ωK).

Here,

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

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

T of

T .

K K

Figure I.3.6. Examples of domains

ωK

Remark I.3.15. The operator I T is called a quasi-interpolation op-erator since it does not interpolate a given function ϕ at the verticesx ∈ N T . In fact, point values are not defined for H 1-functions. Forfunctions with more regularity which are at least in H 2(Ω), the situa-tion is different. For those functions point values do exist and the clas-sical nodal interpolation operator J T : H 2(Ω) ∩ H 1D(Ω) → S 1,0D (T ) canbe defined by the relation (J T (ϕ))(x) = ϕ(x) for all vertices x ∈ N T .

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

ψK = αK

x∈K ∩N T λx ,

αK =

27 if K is a triangle,

256 if K is a tetrahedron,

16 if K is a parallelogram,

64 if K is a parallelepiped.



27/144


0

≤ ψK (x)

≤ 1 for all x

∈ K,

ψK (x) = 0 for all x ∈ K,maxx∈K

ψK (x) = 1.

For every polynomial degree k there are constants cI 1,k andcI 2,k, which only depend on the degree k and the shape pa-rameter CT of T , such that the following inverse estimateshold for all polynomials ϕ of degree k:

cI 1,kϕK ≤ ψ12

K ϕK ,

∇(ψK ϕ)

K

≤ cI 2,kh−1K

ϕ

K .

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

ψE = β E

x∈E ∩N T λx ,

β E = 4 if E is a line segment,

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


0 ≤ ψE (x) ≤ 1 for all x ∈ ωE ,ψE (x) = 0 for all x ∈ ωE ,

maxx∈ωE

ψE (x) = 1.

For every polynomial degree k there are constants cI 3,k,cI 4,k, and cI 5,k, which only depend on the degree k and theshape parameter CT of T , such that the following inverseestimates hold for all polynomials ϕ of degree k:

cI 3,kϕE ≤ ψ12

E ϕE ,∇(ψE ϕ)ωE ≤ cI 4,kh−

12

E ϕE ,ψE ϕωE ≤ cI 5,kh

12

E ϕE .


28/144

28 I. INTRODUCTION

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

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

Figure I.3.7. Examples of domains ωE

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

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


29/144

CHAPTER II

A posteriori error estimates

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

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

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

Solution of auxiliary local problems: On small patches of

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

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

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

Equilibrated residuals: Evaluate approximately a dual varia-

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

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

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

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


30/144

30 II. A POSTERIORI ERROR ESTIMATES

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

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

−∆u = f in Ωu = 0 on ΓD(II.1.1)

∂u

∂n = g on ΓN

in a connected, bounded, polygonal domain Ω

⊂ R2 with boundary Γ

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

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

Find u ∈ H 1D(Ω) such that

(II.1.2) Ω ∇u · ∇v = Ω f v + ΓN gvfor all v ∈ H 1D(Ω).

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

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

Find uT ∈ S 1,0D (T ) such that(II.1.3)

Ω

∇uT · ∇vT =

Ω

f vT +

ΓN

gvT

for all vT ∈ S 1,0D (T ).

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

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

∈ H 1D(Ω) and uT

∈ S 1,0D (

T ) the exact solutions of

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


31/144

II.1. A RESIDUAL ERROR ESTIMATOR 31

Ω ∇(u − uT ) · ∇v = Ω f v + ΓN gv − Ω ∇uT · ∇vfor all v ∈ H 1D(Ω). The right-hand side of this equation implicitlydefines the residual of uT as an element of the dual space of H 1D(Ω).

The Friedrichs and Cauchy-Schwarz inequalities imply for all v ∈H 1D(Ω)

1 1 + c2Ω

v1 ≤ supw∈H 1

D(Ω)w1=1

Ω

∇v · ∇w ≤ v1.

This corresponds to the fact that the bilinear form

H 1D(Ω) v, w →

Ω

∇v · ∇w

defines an isomorphism of H 1D(Ω) onto its dual space. The constantsmultiplying the first and last term in this inequality are related to thenorm of this isomorphism and of its inverse.

The definition of the residual and the above inequality imply theestimate

supw∈H 1

D(Ω)w1=1

Ω

f w + ΓN

gw − Ω

∇uT · ∇w≤ u − uT 1≤

1 + c2Ω supw∈H 1

D(Ω)w1=1

Ω

f w +

ΓN

gw −

Ω

∇uT · ∇w

.

Since the sup-term in this inequality is equivalent to the norm of theresidual in the dual space of H 1D(Ω), we have proved:

The norm in H 1D(Ω) of the error is, up to multiplicativeconstants, bounded from above and from below by the normof the residual in the dual space of H 1D(Ω).

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

II.1.5. Galerkin orthogonality. Since S 1,0D (

T )

⊂ H 1D(Ω), the

error is orthogonal to S 1,0D (T ):


32/144


Ω ∇(u − uT ) · ∇wT = 0for all wT ∈ S 1,0D (T ). Using the definition of the residual, this can bewritten as

Ω

f wT +

ΓN

gwT −

Ω

∇uT · ∇wT = 0

for all wT ∈ S 1,0D (T ). This identity reflects the fact that the discretiza-tion (II.1.3) is consistent and that no additional errors are introducedby numerical integration or by inexact solution of the discrete problem.It is often referred to as Galerkin orthogonality .

II.1.6. L2-representation of the residual. Integration by partselement-wise yields for all w ∈ H 1D(Ω)

Ω

f w +

ΓN

gw −

Ω

∇uT · ∇w

=

Ω

f w +

ΓN

gw −

K ∈T K

∇uT · ∇w

=

Ω

f w +

ΓN

gw +K ∈T

K

∆uT w − ∂K

nK · ∇uT w=K ∈T

K

(f + ∆uT )w +

E ∈E T ,ΓN

E

(g − nE ·∇uT )w

−

E ∈E T ,Ω

E

JE (nE ·∇uT )w.

Here, nK denotes the unit exterior normal to the element K . Note that∆uT

vanishes on all triangles.

For abbreviation, we define element and edge residuals by

RK (uT ) = f + ∆uT

and

RE (uT ) =

−JE (nE · ∇uT ) if E ∈ E T ,Ω,g − nE · ∇uT if E ∈ E T ,ΓN ,0 if E ∈ E T ,ΓD .

Then we obtain the following L2-representation of the residual


33/144


Ω f w + ΓN gw − Ω ∇uT · ∇w=K ∈T

K

RK (uT )w +E ∈E T

E

RE (uT )w.

Together with the Galerkin orthogonality this implies

Ω

f w +

ΓN

gw −

Ω

∇uT · ∇w

= K ∈T K

RK (uT )(w − wT )

+E ∈E T

E

RE (uT )(w − wT )

for all w ∈ H 1D(Ω) and all wT ∈ S 1,0D (T ).

II.1.7. Upper error bound. We fix an arbitrary function w ∈H 1D(Ω) and choose wT = I T w with the quasi-interpolation operator of

section I.3.11 (p. 25). The Cauchy-Schwarz inequality for integrals andthe properties of I T then yield

Ω

f w +

ΓN

gw −

Ω

∇uT · ∇w

=K ∈T

K

RK (uT )(w − I T w) +E ∈E T

E

RE (uT )(w − I T w)

≤K ∈T RK (uT )K w − I T wK + E ∈E T RE (uT )E w − I T wE ≤K ∈T

RK (uT )K cA1hK w1,ωK +E ∈E T

RE (uT )E cA2h1

2

E w1,ωE .

Invoking the Cauchy-Schwarz inequality for sums this gives Ω

f w +

ΓN

gw −

Ω

∇uT · ∇w

≤ max

{cA1, cA2

}K ∈T h2K

RK (uT )

2K


34/144


+ E ∈E T hE RE (uT )2E 12

·

·K ∈T

w21,ωK +E ∈E T

w21,ωE 1

2

.

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

w21,ωK +E ∈E T

w21,ωE 1

2

≤ cw1

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

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

ual, we obtain the following upper bound on the error

u − uT 1 ≤ c∗K ∈T

h2K RK (uT )2K

+

E ∈E T hE RE (uT )2E

12

with

c∗ =

1 + c2Ω max{cA1, cA2}c.The right-hand side of this estimate can be used as an a posteri-

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

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

both approaches are equivalent.


35/144


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

on K

f K = 1

|K | K

f dx

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

gE = 1

|E

| E gdS.

We fix an arbitrary element K and insert the function

wK = (f K + ∆uT )ψK

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

K

RK (uT )wK = K

∇(u − uT ) · ∇wK .

We add

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

K

(f K + ∆uT )2ψK = K

(f K + ∆uT )wK

=

K

∇(u − uT ) · ∇wK − K

(f − f K )wK .

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

K

(f K + ∆uT )2ψK ≥ c2I 1f K + ∆uT 2K

and for the two terms on its right-hand side K

∇(u − uT ) · ∇wK ≤ ∇(u − uT )K ∇wK K ≤ ∇(u − uT )K cI 2h−1K f K + ∆uT K

K

(f − f K )wK ≤ f − f K K wK K ≤ f − f K K f K + ∆uT K .

This proves that

hK f K + ∆uT K ≤ c−2I 1 cI 2∇(u − uT )K + c−

2I 1 hK f − f K K .

(II.1.4)


36/144


Next, we consider an arbitrary interior edge E ∈ E T ,Ω and insert thefunction

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

E

JE (nE · ∇uT )2ψE = E

RE (uT )wE

=

ωE

∇(u − uT ) · ∇wE

−K ∈T E ∈E K

K

RK (uT )wE

= ωE

∇(u − uT ) · ∇wE

−K ∈T E ∈E K

K

(f K + ∆uT )wE

−K ∈T E ∈E K

K

(f − f K )wE

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

E

JE (nE ·∇uT )2ψE ≥ c2I 3JE (nE · ∇uT )2E

and for the three terms on its right-hand side ωE

∇(u − uT ) · ∇wE ≤ ∇(u − uT )1,ωE∇wE 1,ωE≤ ∇(u − uT )1,ωE

· cI 4h−1

2

E JE (nE · ∇uT )E

K ∈T E ∈E K

K

(f K + ∆uT )wE ≤ K ∈T E ∈E K

f K + ∆uT K wE K

≤K ∈T E ∈E K

f K + ∆uT K

· cI 5h1

2


K ∈T E ∈E K K (f − f K )wE ≤

K ∈T E ∈E Kf − f K K wE K


37/144


≤

K ∈T E ∈E Kf − f K K

· cI 5h12


and thus yields

c2I 3JE (nE · ∇uT )E ≤ cI 4h−12

E ∇(u − uT )1,ωE+

K ∈T E ∈E KcI 5h

12

E f K + ∆uT K

+K ∈T E ∈E K

cI 5h1

2

E f − f K K .

Combining this estimate with inequality (II.1.4) we obtain

h12

E JE (nE · ∇uT )E ≤ c−2I 3 cI 5

cI 4 + c−2I 1 cI 2 ∇(u − uT )1,ωE

+ c−2I 3 cI 5 1 + c−2I 1 hE K ∈T E ∈E K

f

−f K

K .

(II.1.5)

Finally, we fix an edge E on the Neumann boundary, denote by K theadjacent element and insert the function

wE = (gE − nE ·∇uT )ψE in L2-representation of the residual. This gives

E

RE (uT )wE = K

∇(u − uT ) · ∇wE − K

RK (uT )wE .

We add E

(gE − g)wE on both sides of this equation and obtain E

(gE − nE · ∇uT )2ψE = E

(gE − nE · ∇uT )wE

=

K

∇(u − uT ) · ∇wE

− K

(f K + ∆uT )wE − K

(f − f K )wE

− E (g − gE )wE .


38/144


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

h1

2

E gE − nE · ∇uT E ≤ c−2I 3 cI 5

cI 4 + c−2I 1 cI 2 ∇(u − uT )K

+ c−2I 3 cI 5

1 + c−2I 1

hK f − f K K + c−2I 3 h

1

2

E g − gE E .

(II.1.6)

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

h2K f K + ∆uT 2K +

1

2

E ∈E K∩E T ,Ω

hE JE (nE · ∇uT )2E

+

E ∈E K∩E T ,ΓN hE gE − nE · ∇uT 2E

12

≤ c∗u − uT 21,ωK+

K ∈T E K∩E K=∅h2K f − f K 21,K

+

E ∈E K∩E T ,ΓN hE g − gE 2E

12

.

The constant c∗ only depends on the shape parameter CT .

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

Denote by u

∈ H 1D(Ω) and u

T ∈ S 1,0D (

T ) the unique solu-

tions of problems (II.1.2) (p. 30) and (II.1.3) (p. 30), re-spectively. For every element K ∈ T define the residual a posteriori error estimator ηR,K by

ηR,K =

h2K f K + ∆uT 2K +

1

2

E ∈E K∩E T ,Ω

hE JE (nE · ∇uT )2E

+

E ∈E K∩E T ,ΓN

hE gE − nE · ∇uT 2E 1

2

,


39/144


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

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

u − uT 1 ≤ c∗K ∈T

η2R,K

+K ∈T

h2K f − f K 2K

+

E ∈E T ,ΓN hE g − gE 2E

12

and

ηR,K ≤ c∗u − uT 21,ωK+K ∈T E K∩E K=∅

h2K f − f K 21,K

+

E ∈E K∩E T ,ΓN

hE g − gE 2E 1

2

hold for all K

∈ T .

Remark II.1.1. The factor 12

multiplying the second term in ηR,K takes into account that each interior edge is counted twice when addingall η2R,K . Note that ∆uT = 0 on all triangles.

Remark II.1.2. The first term in ηR,K is related to the residual of uT with respect to the strong form of the differential equation. Thesecond and third term in ηR,K are related to that boundary operatorwhich is canonically associated with the strong and weak form of thedifferential equation. These boundary terms are crucial when consid-ering low order finite element discretizations as done here. Considere.g. problem (II.1.1) (p. 30) in the unit square (0, 1)2 with Dirichletboundary conditions on the left and bottom part and exact solutionu(x) = x1x2. When using a triangulation consisting of right angledisosceles triangles and evaluating the line integrals by the trapezoidalrule, the solution of problem (II.1.3) (p. 30) satisfies uT (x) = u(x) forall x ∈ N T but uT = u. The second and third term in ηR,K reflectthe fact that uT /∈ H 2(Ω) and that uT does not exactly satisfy theNeumann boundary condition.

Remark II.1.3. The correction terms

hK f − f K K and h1

2E g − gE E


40/144


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

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

◦K 0), and

consider problem (II.1.1) (p. 30) with f = −∆0 and ΓD = Γ. Since K 0

f = − K 0

∆0 = 0

and f = 0 outside K 0, we have

f K = 0

for all K ∈ T . Since

Ωf vT = −

K 0

∆0vT = − K 0

0∆vT = 0

for all vT ∈ S 1,0D (T ), the exact solution of problem (II.1.3) (p. 30) isuT = 0.

Hence, we have

ηR,K = 0

for all K ∈ T , butu − uT 1 = 0.

This effect is not restricted to the particular approximation of f consid-

ered here. Since 0 ∈ C ∞0 ( ◦K 0) is completely arbitrary, we will alwaysencounter similar difficulties as long as we do not evaluate f K ex-actly – which in general is impossible. Obviously, this problem is curedwhen further refining the mesh.

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

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

D

(Ω) bysolving auxiliary problems similar to, but simpler than the original dis-crete problem (II.1.3) (p. 30). Practical considerations and the resultsof the section II.1 suggest that the auxiliary problems should satisfythe following conditions:

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

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

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

should involve as few degrees of freedom as possible.


41/144

I I.2. A CATALOGUE OF ERROR ESTIMATORS 41

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

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

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

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

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

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

V x = span{ϕψK , ρψE , σψE : K ∈ T , x ∈ N K ,E ∈ E T ,Ω, x ∈ N E ,E ∈ E T ,ΓN , E ⊂ ∂ωx,ϕ,ρ,σ

∈ P1

}and

ηD,x = ∇vxωx

where vx ∈ V x is the unique solution of

ωx∇vx · ∇w = K ∈T x∈N K K

f K w + E ∈E T ,ΓN E ⊂∂ωx E gE w

− ωx

∇uT · ∇w

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

ux = uT + vx.

Then

ηD,x = ∇(ux − uT )ωx


42/144


and ux ∈ uT + V x is the unique solution of

ωx

∇ux · ∇w = K ∈T x∈N K

K

f K w + E ∈E T ,ΓN E ⊂∂ωx

E

gE w

for all w ∈ V x. This is a discrete analogue of the following Dirichletproblem

−∆ϕ = f in ωxϕ = uT on ∂ωx\ΓN

∂ϕ

∂n = g on ∂ωx ∩ ΓN .

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

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

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

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

−u

T and that it is comparable to the estimator ηR,T .

Denote by u ∈ H 1D(Ω) and uT ∈ S 1,0D (Ω) the unique solu-tions of problems (II.1.2) (p. 30) and (II.1.3) (p. 30). Thereare constants c N ,1, . . . , c N ,4, which only depend on the shapeparameter CT , such that the estimates

ηD,x ≤ c N ,1

K ∈T x∈N K

η2R,K

12

,

ηR,K ≤ c N ,2 x∈N K\N T ,ΓD η2D,x

12

,

ηD,x ≤ c N ,3u − uT 21,ωx+K ∈T x∈N K

h2K f − f K 2K

+

E ∈E T ,ΓN E ⊂∂ωx

hE g − gE 2E 1

2

,

u

−uT 1 ≤

c N ,4 x∈N T ,Ω∪N T ,ΓN η

2

D,x


43/144


+K ∈T h

2

K f − f K 2

K

+

E ∈E T ,ΓN

hE g − gE 2E 1

2

hold for all x ∈ N T ,Ω ∪N T ,ΓN and all K ∈ T . Here, f K , gE ,and ηR,K are as in sections II.1.8 (p. 35) and II.1.9 (p. 38).

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

∈ N T ,Ω

∪ N T ,ΓN and the corresponding domains ωx

we consider all K ∈ T and the corresponding sets ωK (see figure I.3.2(p. 21)). The considerations from the beginning of this section thenlead to the following definition:Set for all K ∈ T

V K = span{ϕψK , ρψE , σψE : K ∈ T , E K ∩ E K = ∅,E ∈ E K ∩ E T ,Ω,E ∈ E T ,ΓN , E ⊂ ∂ ωK ,ϕ,ρ,σ

∈ P1

}and

ηD,K = ∇vK ωKwhere vK ∈ V K is the unique solution of

ωK ∇vK · ∇w = K ∈T E K∩E K=∅ K f K w + E ∈E T ,ΓN E ⊂∂ωK E

gE w

− ωK

∇uT · ∇w

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

the following Dirichlet problem

−∆ϕ = f in ωK ϕ = uT on ∂ωK \ΓN


44/144


∂ϕ

∂n = g on ∂ωK ∩ ΓN .

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

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

ηD,K ≤ cT ,1

K ∈T E K∩E K=∅η2R,K

1

2

,

ηR,K ≤ cT ,2

K ∈T E K∩E K=∅

η2D,K 1

2

,

ηD,K ≤ cT ,3u − uT 21,ωK+K ∈T E K∩E K=∅

h2K f − f K 2K

+ E ∈E T ,ΓN E ⊂∂ωK

hE

g − gE

2

E 12

,

u − uT 1 ≤ cT ,4K ∈T

η2D,K

+K ∈T

h2K f − f K 2K

+

E ∈E T ,ΓN

hE g − gE 2E 1

2

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

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

V K = span{ϕψK , ρψE : E ∈ E K \E T ,ΓD , ϕ , ρ ∈ P1}


45/144


and

ηN,K = ∇vK K

where vK is the unique solution of K

∇vK · ∇w = K

(f K + ∆uT )w

− 12

E ∈E K∩E T ,Ω

E

JE (nE ·∇uT )w

+ E ∈E K∩E T ,ΓN

E (gE − nE · ∇uT )wfor all w ∈ V K .

Note, that the factor 12

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

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

−∆ϕ = RK (u

T ) in K

∂ϕ

∂n =

1

2RE (uT ) on ∂K ∩ Ω

∂ϕ

∂n = RE (uT ) on ∂K ∩ ΓN

ϕ = 0 on ∂K ∩ ΓD.Again it can be shown that ηN,K also yields upper and lower bounds

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

Denote by u ∈ H 1D(Ω) and uT ∈ S 1,0D (T ) the unique solu-tions of problem (II.1.2) (p. 30) and (II.1.3) (p. 30). Thereare constants cT ,5, . . . , cT ,8, which only depend on the shapeparameter CT , such that the estimates

ηN,K ≤ cT ,5ηR,K ,

ηR,K ≤ cT ,6

K ∈T E K∩E K=∅

η2N,K 1

2

,

ηN,K ≤ cT ,7u − uT 21,ωK+ K ∈T E K∩E K=∅

h2K

f

−f K

2K


46/144


+ E ∈E K∩E T ,ΓN

hE g − gE 2

E 12

,

u − uT 1 ≤ cT ,8K ∈T

η2N,K

+K ∈T

h2K f − f K 2K

+

E ∈E T ,ΓN

hE g − gE 2E 1

2

hold for all K ∈ T

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

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

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

To describe this idea in detail, we consider a finite element spaceY T which satisfies S 1.0D (T ) ⊂ Y T ⊂ H 1D(Ω) and which either consists of higher order elements or corresponds to a refinement of T . We thendenote by wT ∈ Y T the unique solution of

(II.2.1)

Ω

∇wT · ∇vT =

Ω

f vT +

ΓN

gvT

for all vT ∈ Y T .


47/144


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

Ω

∇(wT − uT ) · ∇vT =

Ω

f vT +

ΓN

gvT −

Ω

∇uT · ∇vT

=

Ω

∇(u − uT ) · ∇vT for all vT ∈ Y T , where u ∈ H 1D(Ω) is the unique solution of problem(II.1.2) (p. 30). Since S 1.0D (T ) ⊂ Y T , we may insert vT = wT − uT as a test-function in this equation. The Cauchy-Schwarz inequality for

integrals then implies∇(wT − uT ) ≤ ∇(u − uT ).To prove the converse estimate, we assume that the space Y T satis-

fies a saturation assumption , i.e., there is a constant β with 0 ≤ β


48/144


holds for all vT ∈ S 1,0D (T ), z T ∈ Z T .Now, we write w

T −uT

in the form vT

+ z T

with vT ∈

S 1,0D (

T ) and

z T ∈ Z T . From the strengthened Cauchy-Schwarz inequality we thendeduce that

(1 − γ ){∇vT 2 + ∇z T 2}≤ ∇(wT − uT )2≤ (1 + γ ){∇vT 2 + ∇z T 2}

and in particular

∇z T ≤ 1√ 1

−γ ∇(wT − uT ).(II.2.4)

Denote by z T ∈ Z T the unique solution of (II.2.5)

Ω

∇z T · ∇ζ T =

Ω

f ζ T +

ΓN

gζ T −

Ω

∇uT · ∇ζ T for all ζ T ∈ Z T .

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

Ω

∇z T · ∇ζ T =

Ω

∇(u − uT ) · ∇ζ T (II.2.6)

= Ω ∇(wT − uT ) · ∇ζ T for all ζ T ∈ Z T and

Ω

∇(wT − uT ) · ∇vT = 0(II.2.7)

for all vT ∈ S 1,0D (T ). We insert ζ T = z T in equation (II.2.6). TheCauchy-Schwarz inequality for integrals then yields

∇z T ≤ ∇(u − uT ).On the other hand, we conclude from inequality (II.2.4) and equations

(II.2.6) and (II.2.7) with ζ T = z T that

∇(wT − uT )2 =

Ω

∇(wT − uT ) · ∇(wT − uT )

=

Ω

∇(wT − uT ) · ∇(vT + z T )

=

Ω

∇(wT − uT ) · ∇z T

=

Ω

∇z T · ∇z T

≤ ∇z T ∇z T


49/144


≤ 1√ 1

−γ ∇z T ∇(wT − uT )

and hence

∇(u − uT ) ≤ 11 − β ∇(wT − uT )

≤ 1(1 − β )√ 1 − γ ∇z T .

Thus, we have established the two-sided error bound

∇z T ≤ ∇(u − uT )≤ 1

(1 − β )√ 1 − γ ∇z T .

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

wT since the dimension of Z T is smaller than that of Y T . The com-putation of z T , however, still requires the solution of a global systemand is therefore as expensive as the calculation of uT and wT . But, inmost applications the functions in Z T vanish at the vertices of N T sinceZ T is the hierarchical complement of S

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

lar implies that the stiffness matrix corresponding to Z T

is spectrally

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

More precisely, we assume that there is a bilinear form b on Z T ×Z T which has a diagonal stiffness matrix and which defines an equivalentnorm to ∇· on Z T , i.e.,(II.2.8) λ∇ζ T 2 ≤ b(ζ T , ζ T ) ≤ Λ∇ζ T 2

holds for all ζ T ∈ Z T with constants 0 < λ ≤ Λ.The conditions on b imply that there is a unique function z ∗

T ∈ Z T which satisfies

(II.2.9) b(z ∗T , ζ T ) =

Ω

f ζ T +

ΓN

gζ T −

Ω

∇uT · ∇ζ T

for all ζ T ∈ Z T .The Galerkin orthogonality and equation (II.2.5) imply

b(z ∗T , ζ T ) =

Ω

∇(u − uT ) · ∇ζ T

= Ω ∇z T · ∇ζ T


50/144


for all ζ T ∈ Z T . Inserting ζ T = z T and ζ T = z ∗T in this identity andusing estimate (II.2.8) we infer that

b(z ∗T , z ∗T ) =

Ω

∇(u − uT ) · ∇z ∗T ≤ ∇(u − uT )∇z ∗T ≤ ∇(u − uT ) 1√

λb(z ∗T , z ∗T )

12

and

∇z T 2 = b(z ∗T , z T )

≤ b(z ∗

T , z ∗

T )

12 b(z T , z T )

12

≤ b(z ∗T , z ∗T ) 12√ Λ∇z T .This proves the two-sided error bound√

λb(z ∗T , z ∗T )

12 ≤ ∇(u − uT )

≤√

Λ

(1 − β )√ 1 − γ b(z ∗T , z ∗T )

12 .

We may summarize the results of this section as follows:

Denote by u ∈ H 1D(Ω) and uT ∈ S 1,0D (T ) the unique solu-tions of problems (II.1.2) (p. 30) and (II.1.3) (p. 30), re-spectively. Assume that the space Y T = S

1,0D (T ) ⊕Z T satis-

fies the saturation assumption (II.2.2) and the strengthenedCauchy-Schwarz inequality (II.2.3) and admits a bilinearform b on Z T × Z T which has a diagonal stiffness matrixand which satisfies estimate (II.2.8). Denote by z ∗T ∈ Z T the unique solution of problem (II.2.9) and define the hier-archical a posteriori error estimator ηH by

ηH = b(z ∗T , z ∗T )

12 .

Then the a posteriori error estimates

∇(u − uT ) ≤ √ Λ(1 − β )√ 1 − γ ηH

and

ηH ≤ 1√ λ∇(u − uT )

are valid.

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

sumption and the strengthened Cauchy-Schwarz inequality should be


51/144


uniformly less than 1. Similarly, the quotient Λλ

should be uniformlybounded.

Remark II.2.3. The bilinear form b can often be constructed as fol-lows. The hierarchical complement Z T can be chosen such that itselements vanish at the element vertices N T . Standard scaling argu-ments then imply that on Z T the H 1-semi-norm ∇· is equivalent toa scaled L2-norm. Similarly, one can then prove that the mass-matrixcorresponding to this norm is spectrally equivalent to a lumped mass-matrix. The lumping process in turn corresponds to a suitable numer-ical quadrature. The bilinear form b then is given by the inner-productcorresponding to the weighted L2-norm evaluated with the quadraturerule.

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

Remark II.2.5. The saturation assumption (II.2.2) is used to estab-lish the reliability of the error estimator ηH . One can prove that thereliability of ηH in turn implies the saturation assumption (II.2.2). If the space Y T contains the functions wK and wE of section II.1.8 (p. 35)

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

the form hK f − f K K and h1

2

E g − gE E – the quantity ∇z ∗T ωK isbounded from below by ηR,K for every element K . Together with theresults of section II.1.9 (p. 38) and inequality (II.2.9) this proves – upto the perturbation terms – the reliability of ηH without resorting tothe saturation assumption. In fact, this result may be used to showthat the saturation assumption holds if the right-hand sides f and gof problem (II.1.1) (p. 30) are piecewise constant on T and E T ,ΓN , re-spectively.

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

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

(II.2.10) ∇u − GuT ≤ β ∇u − ∇uT


52/144


holds with a constant 0 ≤ β


53/144


for all vT ∈ V T . Equations (II.2.11) and (II.2.12) imply that

GuT (x) = K ∈T x∈N K

|K ||ωx|∇uT |K

for all x ∈ N T . Thus, GuT may be computed by a local averaging of ∇uT .

We finally set

ηZ,K = GuT − ∇uT K and

ηZ = K ∈T

η2Z,K 12 .One can prove that ηZ yields upper and lower bounds for the error

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

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

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

|v|2 =

Ω

∇v · ∇v

and a quadratic functional J on H 1D(Ω) by

J (v) = 1

2

Ω

∇v · ∇v −

Ω

f v −

ΓN

gv +

Ω

∇uT · ∇v,

where uT ∈ S 1,0D (T ) is the unique solution of problem (II.1.3) (p. 30).The Galerkin orthogonality implies that

J (v) = 1

2

Ω

∇v · ∇v −

Ω

∇(u − uT ) · ∇v

for all v ∈ H 1D(Ω). Hence, J attains its unique minimum at u − uT .Inserting v = u − uT in the definition

adaptive fem

Documents