Element-wise a posteriori estimates based on hierarchical bases for non-linear parabolic problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2005; 63:1146–1173Published online 18 March 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1310

Element-wise a posteriori estimates based on hierarchical basesfor non-linear parabolic problems

Javier de Frutos1,‡ and Julia Novo2,∗,†

1Departamento de Matemática Aplicada, Universidad de Valladolid, Spain2Departamento de Matemáticas, Universidad Autónoma de Madrid, Spain

SUMMARY

We show that the issue of a posteriori estimate the errors in the numerical simulation of non-linearparabolic equations can be reduced to a posteriori estimate the errors in the approximation of anelliptic problem with the right-hand side depending on known data of the problem and the computednumerical solution. A procedure to obtain local error estimates for the p version of the finite elementmethod by solving small discrete elliptic problems with right-hand side the residual of the p-FEMsolution is introduced. The boundary conditions are inherited by those of the space of hierarchical basesto which the error estimator belongs. We prove that the error in the numerical solution can be reducedby adding the estimators that behave as a locally defined correction to the computed approximation.When the error being estimated is that of a elliptic problem constant free local lower bounds areobtained. The local error estimation procedure is applied to non-linear parabolic differential equationsin several space dimensions. Some numerical experiments for both the elliptic and the non-linearparabolic cases are provided. Copyright � 2005 John Wiley & Sons, Ltd.

KEY WORDS: a posteriori error estimation; non-linear parabolic equations; p-version of the finite-element method

1. INTRODUCTION

In its classical form the finite element method (h version) uses piecewise polynomials of fixeddegree p and the mesh size h is decreased for accuracy. In the p version of the finite elementmethod, a fixed mesh is used and p is allowed to be increased. The h–p version combines

∗Correspondence to: J. Novo, Departamento de Matemáticas, Universidad Autónoma de Madrid, Spain.†E-mail: [email protected]‡Research completed while the author was visiting professor at GERAD and HEC Montréal.

Contract/grant sponsor: DGI-MCYT; contract/grant number: BFM2001-2138Contract/grant sponsor: JCYL; contract/grant number: VA044/03

Received 16 March 2004Revised 29 October 2004

Copyright � 2005 John Wiley & Sons, Ltd. Accepted 29 November 2004

ELEMENT-WISE A POSTERIORI ESTIMATES 1147

both approaches. The capability of increasing p coupled with mesh refinement leads to adaptivemethods that can achieve exponential convergence. A posteriori error estimates are a necessaryingredient of such adaptive methods. They are used to estimate the accuracy of computedsolutions and to control adaptive refinement. For the h version of the FEM, many aspects ofa posteriori error estimation and adaptive mesh generation have been discussed in the litera-ture. We refer to the books [1, 2] and the references therein. Comparable literature for the p

version and the h–p version is rather scarce; see for example, References [3–6]—most of thepapers concerning the non-evolutionary case. Some error estimation techniques have also beendeveloped for the parabolic evolutionary case. We refer to Reference [7] where the non-linearparabolic one-dimensional case is considered and to References [8–10] where the authorstreat the parabolic two-dimensional case. The results of References [9, 10] are extended inReference [11] to address the construction of guaranteed computable estimates for fully dis-crete solutions of parabolic problems. Recently, in Reference [12], the estimation of the error ina linear parabolic problem has been reduced to the estimation of the error of some elliptic prob-lems. A similar idea was introduced in Reference [13] to a posteriori estimate the error in the p

version FEM. The procedure is applied to non-linear parabolic differential equations in severalspace dimensions. The error estimation is based on the solution of a global discrete elliptic prob-lem in a space with better approximation capabilities than the space to which the Galerkin solu-tion belongs. The right-hand side of the elliptic problem is the residual of the Galerkin solution.The estimator is proven to be efficient and asymptotically exact and has the property of increas-ing the accuracy of the Galerkin solution when is added to it. To estimate the error in a concreteelement of the partition, the norm of the estimator restricted to this element is computed, butthe procedure to get the estimator is global since the approximation of a global elliptic problemis required.

In this paper we show that the error estimation of a non-linear parabolic problem (both forthe h and p versions of the finite element method) can be reduced to the error estimationof elliptic problems with the right-hand side based on the data and the computed numericalsolution. We extend the idea of Reference [12] to get a global upper bound of the error inthe non-linear parabolic case. For the p version of the finite element method, we obtain localerror indicators based on the solution of suitable defined local problems. If the error in aconcrete element or in a patch of elements is to be estimated, a discrete elliptic problem in theelement or in the patch is solved. The boundary conditions of the problem are homogeneousDirichlet and are inherited by the boundary conditions of the hierarchical bases functions towhich the local indicator belongs. The right-hand side of the problem is the (local) residualof the Galerkin solution. The local estimator maintains the property of improving the Galerkinsolution: the local error in the element or in the patch is shown to decrease when the localindicator is added to the Galerkin solution. The analysis uses some classical ideas borrowedfrom h-FEM hierarchical bases based on a posteriori error estimation theory; see for example,Reference [14].

The outline of the paper is as follows. In Section 2, we state some preliminaries andnotation. We show how to reduce the estimation in the non-linear parabolic case to thatof the elliptic case and extend the techniques of Reference [12]. The rest of the sectionsare developed for p-FEM. Section 3 is devoted to describe the error estimation in a singleelement or in a patch. In the next section, we propose a procedure to estimate the error inthe full domain using the local indicators. Finally, in Section 5, we show some numericalexperiments.

Copyright � 2005 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2005; 63:1146–1173

1148 J. DE FRUTOS AND J. NOVO

2. PRELIMINARIES AND NOTATION

In this paper, we will consider the numerical approximation of non-linear parabolic equationsthat can be written in the form

ut (t, x) − ��u(t, x) + R(u(t, x)) = f (t, x), x ∈ �, 0 � t � T

u(0, x) = u0(x), x ∈ �

u(t, x) = 0, x ∈ ��

(1)

where � ⊂ R2 is a domain with smooth boundary, � is a positive constant, and R is either anon-linear convection term R(u) = (u · ∇)u or a reaction term R(u) = g(u), for some smoothfunction g : R → R, or, of course, a linear combination of reactive and convective terms.The results in the paper are easily extended to other types of non-linear convective terms notnecessarily of quadratic type. Moreover, the case � ⊂ R3 can be treated in a similar way withonly slight modifications.

We will denote by (· , ·) the inner product in L2(�)n and by ‖ · ‖0 the associated norm. Thevalue of n will be 2 for the convection case and 1 for the reaction one. The norm in the Sobolevspace H 1(�)n will be denoted by ‖ · ‖1. It is well known that for a function u ∈ H 1

0 (�)n thenorm ‖u‖1 is equivalent to (∇u, ∇u)1/2. In the sequel, we will use the notation ‖ ·‖1 to denoteany of the two equivalent norms. Finally, H 1

0 (�)n will be the closure of the set of indefinitelydifferentiable functions with compact support respect to the H 1(�)n-norm and H−1(�)n itsdual space.

The standard weak form of problem (1) consists in finding u : (0, T ] → H 10 (�) with u(0) =

u0, such that

(ut , �) + a(u, �) + (R(u), �) = (f, �), ∀� ∈ H 10 (�) (2)

where a(u, v) denotes the Dirichlet bilinear form

a(u, v) = �(∇u, ∇v) u, v ∈ H 10 (�)n (3)

Let X ⊂ H 10 (�)n be a finite dimensional space; we denote by uX : (0, T ] → X the Galerkin

approximation satisfying uX(0, ·) = uX0 ∈ X and

(uXt , �X) + a(uX, �X) + (R(uX), �X) = (f, �X), ∀�X ∈ X (4)

Let us assume that the solution u of (1) belongs to H 10 (�) ∩ Hs(�), s � 2. To fix ideas let

us consider two cases corresponding to the p-FEM and h-FEM approximations. The followingspaces below are indeed the same, the difference being that the partition of � is fixed in thep-FEM and the spaces VN differ one from the other in the degree of the local polynomialswhile the local polynomial degree is fixed in Vh,r , changing the partition of � from one spaceto another.



(1) uX is the p finite element approximation to u.Let us assume for simplicity that � is a rectangular domain and P = {�i}ki=1 is a

conforming partition of � by rectangles, � = ⋃ki=1 �i . Then

X = VN(�) = {vN ∈ (C0(�))n|vN |�i∈ (PN(�i ))

n, ∀�i ∈ P, vN = 0 in ��} (5)

where PN(�i ) is the space of polynomials of degree �N over �i .We denote uN = uX.

(2) uX is the h finite element approximation to u.Let us assume for simplicity that � is a convex polygonal and Th is a shape-regularpartition of �. Then

X = Vh,r (�) = {vh ∈ (C0(�))n|vh|K ∈ (Pr−1(K))n, ∀K ∈ Th, vh = 0 in ��} (6)

where Pr−1(K) is the space of polynomials of degree � r − 1 over K , r � s.We denote uh = uX.

The Galerkin approximation satisfies the optimal a priori bounds

sup0�t�T

‖u(t) − uN(t)‖� � CN−s+�, � = 0, 1 (7)

sup0�t�T

‖u(t) − uh(t)‖� � Chr−�, � = 0, 1 (8)

2.1. Abstract semidiscrete error estimation

In Reference [13] we show that the semidiscrete error estimation of problem (4) can be reducedto the estimation of the error in an abstract elliptic problem. A similar idea has been consideredin Reference [12] to obtain upper a posteriori error bounds in a linear parabolic problem. Theidea in both papers is to introduce an auxiliary infinite-dimensional approximation to the solutionu of (1), called the elliptic reconstruction in Reference [12] and the postprocessed Galerkinapproximation in Reference [13], see also References [15–17]. We start the analysis obtainingglobal a posteriori upper bounds of the error in the non-linear parabolic case.

For each t∗ � T , we define the following elliptic problem: find U(t∗) ∈ H 10 (�)n such that

a(U(t∗), �) = (gX(t∗), �), ∀� ∈ H 10 (�)n (9)

where the right-hand side gX is defined in variational form by

(gX(t∗), �) = −(R(uX(t∗)), �) − (uXt (t∗), �) + (f (t∗), �), ∀� ∈ H 1

0 (�)n (10)

We remark that although defined for each t∗ � T , the postprocessed Galerkin approximation(elliptic reconstruction) U(·) is not the solution of an evolutionary problem but of a collectionof independent elliptic problems in which time appears merely as a parameter. On the otherhand, for each t∗ � T the discrete approximation uX(t∗) defined by (4) can be seen as the



Galerkin solution in X to (9). The error E(t∗) = U(t∗) − uX(t∗) satisfies the equation

a(E(t∗), �) = −a(uX(t∗), �) − (R(uX(t∗)), �) − (uXt (t∗), �) + (f (t∗), �), ∀� ∈ H 1

0 (�)n

(11)

where the right-hand side depends only on the computed numerical solution, uX(t∗), and knowndata of the problem, f (t∗). Indeed, it is the residual of the Galerkin solution with time frozenat level t∗.

Let us denote by w the solution of the elliptic problem

a(w, v) = (g, v), ∀v ∈ H 10 (�)n (12)

and let wX ∈ X be the corresponding Galerkin approximation. Following [12], we assume thatthere exist a posteriori estimator functions E = E(wX, g, L2) and E = E(wX, g, H−1) inL2(�)n and H−1(�)n, respectively, such that the following upper bounds hold:

‖w − wX‖0 �E(wX, g, L2), ‖w − wX‖−1 �E(wX, g, H−1) (13)

The next theorem establishes an upper bound for the error, analogous to the one obtained inReference [12, Theorem 3.1] in the linear case.

Theorem 1Let u be the solution of (1) and uX its Galerkin approximation. Then, the following a posteriorierror bounds hold for 0 < t � T .

‖(u − uX)(t)‖0 � eKt‖u0 − uX0 ‖0 + eKtE(uX(0), gX(0), L2) + E(uX(t), gX(t), L2)

+CeKt t1/2[

max0�s�t

E(uX, gX, L2) + max0�s�t

E(uXt , gX

t , H−1)

](14)

ProofLet A : H 1

0 (�)n → H−1(�)n be the operator defined by

(Au, v) = a(u, v), ∀u, v ∈ H 10 (�)n (15)

It follows that � = u − U satisfies the equation

�(t) = e−At�(0) +∫ t

0e−A(t−s)(R(U) − R(u)) ds

+∫ t

0e−A(t−s)(R(uX) − R(U)) ds +

∫ t

0e−A(t−s)(uX

t − Ut) ds (16)

Taking into account [16, Lemma 4.1] that

‖e−Atv‖0 �C‖v‖−1√

t, v ∈ L2(�)n, t > 0 (17)



and since e−At is a contraction we get

‖�(t)‖0 � ‖�(0)‖0 + C

∫ t

0

‖R(U) − R(u)‖−1√t − s

ds

+C

∫ t

0

‖R(uX) − R(U)‖−1√t − s

ds + C

∫ t

0

‖uXt − Ut‖−1√

t − sds (18)

Using now that (see References [17], [18])

‖R(U) − R(u)‖−1 � K(u)‖U − u‖0 (19)

‖R(uX) − R(U)‖−1 � K(u)‖uX − U‖0 (20)

we obtain

‖�(t)‖0 � ‖�(0)‖0 + CK

∫ t

0

‖�(t)‖0√t − s

ds

+CK

∫ t

0

‖uX − U‖0√t − s

ds + C

∫ t

0

‖uXt − Ut‖−1√

t − sds (21)

And then,

‖�(t)‖0 � ‖�(0)‖0 + CK

∫ t

0

‖�(t)‖0√t − s

ds

+CKt1/2 max0�s�t

E(uX, gX, L2) + CKt1/2 max0�s�t

E(uXt , gX

t , H−1) (22)

A standard application of the generalized Gronwall lemma (see Reference [19]) gives

‖�(t)‖0 � eKt‖�(0)‖0 + Ct1/2eKt

(max

0�s�tE(uX, gX, L2) + max

0�s�tE(uX

t , gXt , H−1)

)(23)

Now, adding and subtracting U to bound (u−uX) as is done in Reference [12, Theorem 3.1],we conclude the proof. �

Using the same proof, a similar bound for the H 1 norm of the error can be obtainedchanging only E(uX

t , gXt , L2) by E(uX

t , gXt , H 1) and E(uX

t , gXt , H−1) by E(uX

t , gXt , L2).

We observe that Theorem 1 allows to get an upper global a posteriori error bound for theerror of the non-linear parabolic problem using only upper global a posteriori error boundsof elliptic problems that depend on the data and the computed solution uX. However, theestimation of the error at a time t requires the estimation of the error of a family of ellipticproblems the right-hand side of which depends on s, for all s ∈ [0, t].

A simpler procedure to estimate the error in the Galerkin approximation to (1) at time t∗ is totake into account that uX(t∗) is also the Galerkin approximation in X to the solution of (9), and,



under the regularity assumptions specified below, E(T ∗) = U(T ∗)−uX(T ∗) is an asymptoticallyexact error estimator of u(T ∗) − uX(T ∗). In consequence, each error estimator for the ellipticproblem (9) can be used to a posteriori estimate the error in the parabolic problem at fixedtimes t∗. Furthermore, each asymptotically exact error estimator for (9) becomes automaticallyan asymptotically exact error estimator for (1).

In what follows, we will assume that the forcing term, f , and the initial data u0 in (1) aresuch that u, ut ∈ L∞ ([0, T ], H s(�)n), s � 2. We denote this by

Ki = sup0�t�T

∥∥∥∥diu

dt i

∥∥∥∥s

, i = 0, 1 (24)

We remark that this kind of bound essentially requires some compatibility conditions on thedata of the problem; see for example, Reference [20].

Under hypothesis (24) it can be proven that the solution of (9), U(t∗), is in fact an improvedapproximation to u(t∗); see References [15–18]. More precisely, the following bounds hold:

‖u(t∗) − U(t∗)‖� � C log(N)N−s−1+�, X = VN, � = 0, 1 (25)

‖u(t∗) − U(t∗)‖� � C| log(h)|hr+1−�, X = Vh,r , � = 0, 1 (26)

To prove that E(t∗) is an asymptotically exact estimator of the semidiscrete error u(t∗) −uX(t∗) we assume that bounds (7) and (8) are optimal (i.e. the error is exactly O(N−s+�) inthe p finite element case and O(hr−�) in the h finite element case). It is easy to see that

‖E(t∗)‖� − ‖u(t∗) − U(t∗)‖� � ‖u(t∗) − uX(t∗)‖�

‖E(T ∗)‖� + ‖u(t∗) − U(T ∗)‖� � ‖u(t∗) − uX(T ∗)‖�

(27)

Using then (25) and (26), the conclusion is reached.The procedure we propose, see Reference [13], to estimate the error in the p finite element

approximation reads as follows. We choose M > N and compute UMN (t∗) ∈ VM(�) satisfying

a(UMN (t∗), �M) = −(R(uN(t∗), �M) − (uN

t (t∗), �M) + (f (t∗), �M) (28)

∀�M ∈ VM(�). This new computable approximation is also called the postprocessed approxi-mation. The difference, UM

N (t∗) − uN(t∗), is then used to estimate the error in uN(t∗). Thesame idea can be used in the h version. In this case the postprocessed approximation belongsto the space Vh,r+1 of local polynomials of degree � r + 1, or to the space Vh′,r , h′ < h, oflocal polynomials of the same degree r as the Galerkin solution but over a finer mesh; seeReferences [17, 18].

The postprocessed approximation provides an efficient estimation of the error in the non-linearparabolic problem being itself an improved approximation to this problem (see Reference [13]).However, a global discrete elliptic problem must be solved at every time t∗ where the errorwants to be estimated. In the next section, we show a procedure to estimate local errorsby solving local problems. Taking into account the comments made in this section we willconcentrate on the estimation of the error in an elliptic problem. We consider in the sequelonly the p version of the finite element method.



3. LOCAL ERROR ESTIMATION IN A SINGLE ELEMENT OR IN A PATCH

In this section, we propose a procedure to estimate the error in a single element of the partitionor in an arbitrary patch of elements. Although the procedure is applicable to both the h or p

version of the finite element method, for simplicity in the exposition we restrict ourselves tothe p version in which the hierarchical bases approach we follow appears as the natural wayto practically implement the Galerkin method.

As a model problem we consider the elliptic boundary value problem in variational form:

a(u, v) = (f, v), ∀v ∈ H 10 (�) (29)

where we can suppose that � = 1.Our purpose is to estimate the error in the Galerkin approximation of (29), uN ∈ VN(�)

satisfying

a(uN, vN) = (f, vN), ∀vN ∈ VN(�) (30)

The definition of the local estimator we propose is closely related to the hierarchical basesfunctions of the space VN(�). For the sake of completeness we introduce this basis on thereference square Q = [−1, 1] × [−1, 1]. The hierarchical basis functions of the p version areconstructed as tensorial products of the one-dimensional functions:

�0(�) = 12 (1 − �), −1 � � � 1

�1(�) = 12 (1 + �), −1 � � � 1

�j (�) = cj

∫ �

−1Lj−1(�) d�, −1 � � � 1, 2 � j �N

where Lk is the kth Legendre polynomial and ck is a normalization factor. In two dimensions,the basis functions consists of three kind of functions (see for example, Reference [21] for acomplete description):

1. Four nodal basis functions: they are bilinear functions associated with each vertex of thereference square,

�n1(�1, �2) = �0(�1)�0(�2), �n2

(�1, �2) = �1(�1)�0(�2)

�n3(�1, �2) = �1(�1)�1(�2), �n4

(�1, �2) = �0(�1)�1(�2)

2. 4(N − 1) edge basis functions: Each basis function is associated with one side and iszero on the other three sides. For 2 � j �N they are

�e1,j(�1, �2) = �j (�1)�0(�2), �e2,j

(�1, �2) = �1(�1)�j (�2)

�e3,j(�1, �2) = �j (�1)�1(�2), �e4,j

(�1, �2) = �0(�1)�j (�2)

3. (N − 1)2 interior basis functions: They are bubble functions that vanish on the four sidesof the standard square:

�j,k(�1, �2) = �j (�1)�k(�2), 2 � j, k �N



01

23

45

6

01

23

45

60

0.2

0.4

0.6

0.8

1

01

23

45

6

01

23

45

60

0.2

0.4

0.6

0.8

1

01

23

45

6

01

23

45

60

0.2

0.4

0.6

0.8

1

Figure 1. Some of the nodal, side and interior bases functions for N = 2.

After assembling, the global basis functions can still be classified into nodal, edge andinterior functions. Each nodal function is associated to a node of the partition and is supportedat the four elements that share this vertex. Each side function is associated to one side of thepartition. There are (N − 1) for each side and they are not identically zero only at the twoelements that share the side. Finally, the interior functions are supported only at one element;there are (N − 1)2 for each element in the partition. In Figure 1, we have represented someof the bases functions for N = 2 in a 9 element partition of � = [0, 6] × [0, 6].

Let us suppose that the error u − uN is to be estimated at one element or at a particularpatch of elements. We will denote the selected element or patch by � ⊂ �. Let M > N , thespace VM(�) (defined as in (6) changing � by �) is the subspace of VM(�) generated by thebasis functions in VM(�) that are identically zero out of �, vanishing at the boundary.

The local error estimator we propose is obtained by solving the following problem in �:find eM

N ∈ VM(�) such that

a(eMN , vM) = (f, vM) − a(uN, vM) ∀vM ∈ VM(�) (31)

Note that the right-hand side in (31) is the residual of the Galerkin approximation. In thesequel we will use the notation

(rN , v) = (f, v) − a(uN, v) = a(u − uN, v), v ∈ H 10 (�) (32)



for the Galerkin residual in weak form. The boundary conditions in (31) are those satisfied bythe functions of VM(�), i.e. homogeneous Dirichlet in �. Let us observe that, for example,to estimate the error in an element, we only use the interior bases functions associated tothat element. To estimate the error in a star (the patch formed by the support of a nodalbasis function) we use the nodal function that defines the star, the side functions associatedto the interior sides of the star and the interior functions of the elements in the star. Thesolution of local Dirichlet problems on stars to estimate the error in the h-FEM was proposedin Reference [22]. Recently in References [23, 24], the performance of some a posteriori errorestimators obtained solving local problems on stars has been studied.

Let us decompose the Galerkin space VN(�) into two subspaces

VN(�) = X ⊕ X (33)

where X = VN(�) is the space generated by the basis functions that are identically zero out of� and X is generated by the remainder bases functions. Let Y be the orthogonal complementwith respect to the inner product in H 1

0 (�) (a(·, ·)) of X into VM(�)

VM(�) = X ⊕ Y (34)

We define the enhanced space VM

N (�)

VM

N (�) = VN(�) ⊕ Y = X ⊕ VM(�) (35)

The functions in VM

N (�) are piecewise polynomials of degree less or equal to M in � anddegree less or equal to N out of �. Note that when we add the estimator to the Galerkinsolution, we obtain a new solution

uMN = uN + eM

N (36)

that belongs to the enhanced space VM

N (�). We will prove that the new solution uMN is an

improved approximation to the solution u in (29). This approximation cannot be better than

the best approximation in VM

N (�), that is, the Galerkin approximation uMN ∈ V

M

N (�) obtainedby solving the global problem

a(uMN , vM

N ) = (f, vMN ) ∀vM

N ∈ VM

N (�) (37)

Let us denote by

eMN = uM

N − uN (38)

The estimation we propose is obtained by taking the difference between two solutions ofproblem (29) with different accuracy (eM

N = uMN − uN). What we prove below is that our

estimator is efficient, in the sense of being equivalent to the estimator eMN (obtained as the

difference between the best approximation in VM

N (�) and uN ). We observe that our estimatoris obtained solving a locally defined problem, while to compute eM

N one should solve a globalproblem. Note also that the new solution uM

N belongs to a space in which the degree of thelocal polynomials used has been increased, with respect to the solution being estimated, onlyat � and, then, the estimator eM

N is hoped to be a good indicator of the local error of uN in �.



In the following lemma, we prove that the estimator eMN defined in (31) belongs to the

orthogonal complement Y .

Lemma 1The estimator eM

N belongs to the space Y and solves

a(eMN , v

Y) = (rN , v

Y) ∀v

Y∈ Y (39)

ProofTaking into account that eM

N ∈ VM(�) and using (34) we can decompose

eMN = e

X+ e

Y, e

X∈ X, e

Y∈ Y (40)

Then, using the definition of eMN (31) we have

a(eX, v

X) + a(e

Y, v

X) = (rN , v

X) ∀v

X∈ X

a(eX, v

Y) + a(e

Y, v

Y) = (rN , v

Y) ∀v

Y∈ Y

(41)

Now, taking into account that the spaces X and Y are orthogonal the above equations can bereduced to

a(eX, v

X) = (rN , v

X) ∀v

X∈ X

a(eY, v

Y) = (rN , v

Y) ∀v

Y∈ Y

(42)

The right-hand side in the first equation vanishes, owing to uN being the Galerkin approximationin VN(�) and X ⊂ VN(�) (Galerkin orthogonality). Then e

X= 0, which concludes the proof.

�

In view of Lemma 1 there are two ways to calculate the local estimator eMN in practice. One

can compute the estimator using the definition (31) or compute the complement Y explicitlyand then solve a discrete elliptic problem in this complement (39).

The following lemma will be used to get a local lower bound for the estimator.

Lemma 2The following relation holds:

a(eMN , eM

N ) = a(eMN , eM

N ) (43)

ProofFrom the definition of eM

N , we have a(eMN , eM

N ) = (rN , eMN ). For the estimator eM

N , we have

a(eMN , eM

N ) = a(uMN , eM

N ) − a(uN, eMN )

= (f, eMN ) − a(uN, eM

N ) = (rN , eMN ) (44)

where we have used the fact that uMN is the Galerkin approximation in V

M

N (�) and eMN ∈

Y ⊂ VM

N (�). �



Theorem 2The following local lower bound is satisfied:

‖eMN ‖1, � � ‖eM

N ‖1, � (45)

ProofNote that

‖eMN ‖2

1, �= a(eM

N , eMN ) = a(eM

N , eMN ) =

∫�

∇ eMN ∇ eM

N dx � ‖eMN ‖1, �‖eM

N ‖1, �

after using Lemma 2 and Cauchy–Schwarz inequality. �

The equivalence between our local error estimator and eMN will depend on the loss incurred

by neglecting the coupling terms in the Galerkin problem that solves uMN . For example, in

the case of � being a single element, � = �i , we do not consider the interaction betweenthe interior functions of �i and the side and nodal functions that have support containing �i .What we avoid, calculating the local estimator eM

N in the orthogonal complement Y , is theinteraction between the added interior bases functions (up to degree M > N) and the interiorbases functions in the Galerkin solution (those of degree less or equal to N ). To measurethe coupling between spaces, we will use the constant in the strengthened Cauchy–Schwarzinequality (see References [2, 14, 25]). For X and Y , being finite-dimensional subspaces ofH 1

0 (�) satisfying X ∩ Y = {0}, there exists a constant � ∈ [0, 1) such that

|a(vX, vY )| � �‖vX‖1‖vY ‖1 ∀vX ∈ X, vY ∈ Y (46)

Reference [25].The following theorem in addition to Theorem 2 gives the equivalence between our estimator

and eMN . The proof is basically the same as the one appearing in Reference [14] in the context

of the h-FEM.

Theorem 3Let � < 1 be the constant in the strengthened Cauchy–Schwarz inequality between the spacesVN(�) and Y (that is, taking in (46) X = VN(�) and Y = Y ). Then, the following boundholds:

‖eMN ‖1 � 1√

1 − �2‖eM

N ‖1 (47)

As we mentioned before, from Theorems 2 and 3 (taking into account that ‖eMN ‖1, � =

‖eMN ‖1), we obtain the equivalence between our local estimator and eM

N

‖eMN ‖1 � ‖eM

N ‖1 � 1√1 − �2

‖eMN ‖1 (48)

We observe that, in the definition of the estimator eMN , we could use any space Y satisfying

VM

N (�) = VN(�) ⊕ Y and the results of the section would still be valid, changing only thevalue of �. The choice of Y reduces the size of �, which leads to a tight equivalence betweeneMN and eM

N (48). We next show an example to check this fact.



Table I. Computed values of �. Poisson problem.

N M = N + 2 M = N + 4 M = N + 6

4 0.3213 0.3414 0.34406 0.3002 0.3320 0.33938 0.2881 0.3271 0.3389

10 0.2802 0.3237 0.339212 0.2747 0.3207 0.3391

We choose � to be [−1, 3] × [−1, 1] and consider a uniform partition of � into ninerectangles denoted by

�i,j = [−1 + 43 (j − 1), −1 + 4

3 j ] × [−1 + 23 (i − 1), −1 + 2

3 i], i, j = 1, 2, 3

Then we take � = �2,3. The Galerkin space is based on local polynomials of degree less orequal to N at the nine elements. The space Y is the orthogonal complement (respect to theinner product in H 1

0 ) of the space of interior functions in �2,3 up to degree N into the spaceof interior functions in �2,3 up to degree M . In Table I, we have represented the numericallycomputed values of � in (46) (X = VN(�), Y = Y ) for different values of N and M .

We can observe in Table I that all the values of � are quite small. The size of � slightlyincreases with M and decreases with N . This fact implies that the procedure we propose toestimate the local errors does not deteriorate when the degree of the local polynomials isincreased in the Galerkin solution we are estimating. Note that for a value of � = 0.35, whichis bigger than all the values appearing in the table, the relation between the estimators eM

N andeMN using (48) is the following:

‖eMN ‖1 � ‖eM

N ‖ � 1.07 ‖eMN ‖1 (49)

so that both indicators give almost the same estimations for all the values of N and M inTable I. As a consequence, to estimate the local error in the subdomain �2,3, in this example,we can obtain essentially the same results using the two different procedures:

1. Calculate another global Galerkin solution, increasing the degree of the local polynomialsin �2,3 (up to degree M) and maintaining degree N in the rest of the elements (eM

N ).2. Solve a local problem in �2,3 with homogeneous Dirichlet boundary conditions, using

polynomials of degree M (eMN ).

The number of degrees of freedom needed in the first procedure in this simple example is(4 + 4(N − 1)(2N + 1) + (M − 1)2), while the number of degrees of freedom in the secondis just (M − 1)2, if we do not compute explicitly the complement Y and solve (31), or((M − 1)2 − (N − 1)2) if we calculate the complement and solve (39).

While the theory has been developed for elliptic problems associated with the Laplaceoperator, it is possible to extend the results to more complicated cases. We next consider anexample borrowed from the linear elasticity theory. The equations of linear elasticity for plane



Table II. Computed values of �. Linear elasticity problem.

� = 0.30 � = 0.49

N M = N + 2 M = N + 4 M = N + 6 M = N + 2 M = N + 4 M = N + 6

4 0.4966 0.5464 0.5569 0.7286 0.8005 0.82046 0.4590 0.5246 0.5454 0.6730 0.7684 0.80218 0.4313 0.5048 0.5461 0.6334 0.7369 0.7814

10 0.4108 0.4877 0.5219 0.6151 0.7074 0.760212 0.3957 0.4732 0.5110 0.6053 0.6804 0.7394

strain can be written in the form

−E(1 − �2)

1 − 2�

(�2

u

�x2+ 1 − 2�

2(1 − �)

�2u

�y2+ 1

2(1 − �)

�2v

�x�y

)= Fx

−E(1 − �2)

1 − 2�

(1 − 2�

2(1 − �)

�2v

�x2+ �2

v

�y2+ 1

2(1 − �)

�2u

�x�y

)= Fy

(50)

where u and v denote the displacement in the x and y directions, E is Young’s modulus and� is Poisson’s ratio.

In Table II we show the values of the strengthened Cauchy–Schwarz inequality constant �computed with respect to the bilinear form associated with system (50), for the same spacesX and Y as in the previous example. As we can see, for a fixed value of Poisson’s ratio thebehaviour of constant � is similar to the Laplace operator case in the sense that it increaseswith M and decreases with N . As might be expected, the value of � deteriorates when �approaches 1/2 (nearly incompressible materials). However, for � = 0.49 and M = N + 2, theworst computed value gives a bound,

‖eMN ‖1 � ‖eM

N ‖ � 1.46 ‖eMN ‖1 (51)

which is still tight enough for error estimation purposes.In the next theorem, we prove that the Galerkin error can be locally reduced adding the

local error estimator eMN to the Galerkin solution uN .

Theorem 4The following relation is satisfied:

‖u − (uN + eMN )‖2

1, �= ‖u − uN‖2

1, �− ‖eM

N ‖21, �

(52)

ProofWe first observe that

‖u − (uN + eMN )‖2

1 = a(uN − u + eMN , uN − u + eM

N )

= ‖u − uN‖21 + ‖eM

N ‖21 − 2a(u − uN, eM

N )

= ‖u − uN‖21 + ‖eM

N ‖21 − 2(rN , eM

N )

= ‖u − uN‖21 + ‖eM

N ‖21 − 2a(eM

N , eMN )

= ‖u − uN‖21 − ‖eM

N ‖21



after using the definition of eMN (31). Then, taking into account that the local estimator is

identically zero out of �, the conclusion is reached. �

From the proof of Theorem 4, we observe that the square of the norm of the global errorin the solution uN + eM

N is reduced in the same quantity as the square of the norm of thelocal error. Note that we always get an improved solution adding the local estimator (unlessthe indicator eM

N = 0). The improvement obtained in the new solution heavily relies on thesize of the norm of the error indicator.

Another important consequence is deduced from (52). Taking into account that ‖u − (uN +eMN )‖1, � � 0, the following local lower error bound between our error estimator and the Galerkin

error is obtained

‖eMN ‖1, � � ‖u − uN‖1, � (53)

The constant free lower bound (53) implies the reliability of our error indicator since the sizeof eM

N dictates the need of refinement of the mesh (or increase of the degree of the local

polynomials). In the particular case of � being a single element �i bound (53) becomes

‖eMN ‖1, i � ‖u − uN‖1, i (54)

pointing out that the error in the element �i exceeds a prescribed accuracy, whenever the sizeof the norm of the local estimator does.

4. ESTIMATION OF THE ERROR IN THE FULL DOMAIN

In this section, we propose a procedure to estimate the error in the full domain �. The estimatorof the global error is obtained as a combination of local error indicators that are obtained bysolving finite-dimensional local problems. Adding these indicators to the Galerkin solution weget a reduction of the error over the full domain �.

We recall that a conforming partition of � into k subrectangles has been considered P ={�i}ki=1. We will denote by l the number of interior edges in the partition P arranged in anarbitrary fashion. The procedure we propose reads as follows:

1. We first choose M > N .2. Then, we compute the local error indicators corresponding to every element of the par-

tition. That is, for i = 1, . . . , k, we set � = �i , and compute Einti ∈ VM(�i ) such

that

a(Einti , vM) = (rN , vM) ∀vM ∈ VM(�i ) (55)

We observe that (55) can also be solved in the orthogonal complement of VN(�i ) intoVM(�i ) (see (39)). The computation of these local interior indicators can be carried outusing a parallel execution since all the problems are completely independent. We defineEint = ∑k

i=1 Einti .

3. To take into account the error along the interior edges of the partition P, we computelocal error indicators based on the side functions. Let us denote by �j , j = 1, . . . , l, the

interelement edges of the interior of the mesh. We write V�j

N for the subspace of VN(�)



generated by the N − 1 side bases functions associated to the interior edge �j . Then, wehave the hierarchical decomposition

V�j

M = V�j

N ⊕ W �j (56)

where W �j is the subspace that corresponds to the span of the additional side basesfunctions. Now, for j = 1, . . . , l, we compute the local edge indicator Eed

j ∈ W �j satisfying

a(Eedj , wj ) = (f, wj ) − a

(uN + Eint +

j−1∑m=1

Eedm , wj

)∀wj ∈ W �j (57)

Each discrete elliptic problem (57) is solved in the finite-dimensional space W �j ofdimension M − N . The right-hand side is the residual of the modified solution that isobtained adding to the Galerkin solution all the local indicators that have been previouslycomputed. We set Eed = ∑l

m=1 Eedm .

In the next theorem, we prove that adding the local interior and side indicators to theGalerkin solution reduces the global error.

Theorem 5The following relation holds:

‖u − (uN + Eint + Eed)‖21 = ‖u − uN‖2

1 −k∑

i=1‖Eint

i ‖21 −

l∑j=1

‖Eedj ‖2

1 (58)

ProofReasoning as in the proof of Theorem 4 changing eM

N by Eint, we get

‖u − (uN + Eint)‖21 = ‖u − uN‖2

1 − ‖Eint‖21 = ‖u − uN‖2

1 −k∑

i=1‖Eint

i ‖21 (59)

where in the last identity we have used that the local interior indicators are orthogonal. Toconclude the proof we proceed by induction. For j = 1, reasoning again as in Theorem 4 wehave

‖u − (uN + Eint + Eed1 )‖2

1 = a(uN + Eint − u + Eed1 , uN + Eint − u + Eed

1 )

= ‖u − (uN + Eint)‖21 + ‖Eed

1 ‖21 − 2a(u − (uN + Eint),Eed

1 )

= ‖u − (uN + Eint)‖21 + ‖Eed

1 ‖21 − 2(f,Eed

1 ) − 2a(uN + Eint,Eed1 )

= ‖u − (uN + Eint)‖21 + ‖Eed

1 ‖21 − 2a(Eed

1 ,Eed1 )

= ‖u − (uN + Eint)‖21 − ‖Eed

1 ‖21

= ‖u − uN‖21 −

k∑i=1

‖Einti ‖2

1 − ‖Eed1 ‖2

1



where we have used the definition of Eed1 (57) and (59). Let us now suppose that

∥∥∥∥∥u −(

uN + Eint +m−1∑j=1

Eedj

)∥∥∥∥∥2

1

= ‖u − uN‖21 −

k∑i=1

‖Einti ‖2

1 −m−1∑j=1

‖Eedj ‖2

1

holds true for m � l, and let us prove

∥∥∥∥∥u −(

uN + Eint +m∑

j=1Eed

j

)∥∥∥∥∥2

1

= ‖u − uN‖21 −

k∑i=1

‖Einti ‖2

1 −m∑

j=1‖Eed

j ‖21 (60)

to conclude.Reasoning as before, we have

∥∥∥∥∥u −(

uN + Eint +m∑

j=1Eed

)∥∥∥∥∥2

1

= a

(uN + Eint +

m−1∑j=1

Eedj − u + Eed

m , uN + Eint +m−1∑j=1

Eedj − u + Eed

m

)

=∥∥∥∥∥u −

(uN + Eint +

m−1∑j=1

Eedj

)∥∥∥∥∥2

1

+ ‖Eedm ‖2

1

−2a

(u −

(uN + Eint +

m−1∑j=1

Eedj

),Eed

m

)

Then

∥∥∥∥∥u −(

uN + Eint +m∑

j=1Eed

)∥∥∥∥∥2

1

=∥∥∥∥∥u −

(uN + Eint +

m−1∑j=1

Eedj

)∥∥∥∥∥2

1

+ ‖Eedm ‖2

1

−2(f,Eedm ) − 2a

(uN + Eint +

m−1∑j=1

Eedj ,Eed

m

)

=∥∥∥∥∥u −

(uN + Eint +

m−1∑j=1

Eedj

)∥∥∥∥∥2

1

+ ‖Eedm ‖2

1 − 2a(Eedm ,Eed

m )



And, finally,

∥∥∥∥∥u −(

uN + Eint +m∑

j=1Eed

)∥∥∥∥∥2

1

=∥∥∥∥∥u −

(uN + Eint +

m−1∑j=1

Eedj

)∥∥∥∥∥2

1

− ‖Eedm ‖2

1

= ‖u − uN‖21 −

k∑i=1

‖Einti ‖2

1 −m∑

j=1‖Eed

j ‖21

which finishes the proof. �

We deduce from Theorem 5 that the global error in the new solution is always reducedwhenever any of the local interior or local edge indicators are not identically zero. As inTheorem 4, we know the exact quantity in which the square of the norm of the errordiminishes.

5. NUMERICAL EXPERIMENTS

5.1. Elliptic problems: smooth solutions

In this section we present some numerical experiments to illustrate the behaviour of our localindicators. We consider first an elliptic problem with smooth solution: find u ∈ H 1

0 (�) suchthat

a(u, v) = (f, v) ∀v ∈ H 10 (�) (61)

with � = [−2, 2] × [−2, 2] and right-hand side f defined appropriately so that u = (4 − x2)

(4 − y2) arctan(1 − x2 − y2) is the exact solution. This solution exhibits a mild interior layeras can be seen in Figure 2. For the Galerkin solution, we consider a fixed uniform mesh withnine elements and the same degree N for the local polynomials at every element. We denotethe elements by

�i,j = [−2 + 43 (j − 1), −2 + 4

3j ] × [−2 + 43 (i − 1), −2 + 4

3 i], i, j = 1, 2, 3

Table III shows, for each one of the subdomains �ij , the error in the finite element approxi-mation ‖u − uN‖1,�ij

, the interior error estimator ‖Eint‖1,�ijand the interior plus edge error

estimator ‖Eint + Eed‖1,�ij. They appear as first, second and third data in each cell labelled

with the index of the corresponding subdomain. We have also included in the first row of thetable the global energy norm of the same data. The results correspond to the value N = 4while the error estimator has been computed with M = 6; see Sections 3 and 4.

We can check in the table that the interior estimator gives a lower bound for the error, i.e.‖Eint‖1 � ‖u−uN‖1. Moreover, every local interior indicator gives a local lower bound for theerror at the element in which the indicator has been computed, ‖Eint‖1,�ij

� ‖u − uN‖1,�ij,



-2

-1

0

1

2

-2

-1

0

1

2-10

-5

0

5

10

Figure 2. Graph of the exact solution of the elliptic problem.

Table III. Galerkin errors, interior estimators and interiorplus edge estimators. N = 4, M = 6.

‖u − uN‖1 = 4.7279e − 2‖Eint‖1 = 4.0522e − 2

‖Eint + Eed‖1 = 4.6397e − 2

j = 1 j = 2 j = 3

9.5529e − 3 2.1607e − 2 9.5529e − 3i = 3 6.6722e − 3 1.9122e − 2 6.6722e − 3

8.7997e − 3 2.1444e − 2 8.4882e − 3

2.1607e − 2 1.6727e − 3 2.1607e − 2i = 2 1.9122e − 2 1.1357e − 3 1.9122e − 2

2.1446e − 2 1.8847e − 3 2.1444e − 2

9.5529e − 3 2.1607e − 2 9.5529e − 3i = 1 6.6722e − 3 1.9122e − 2 6.6722e − 3

9.1002e − 3 2.1446e − 2 8.7997e − 3

i, j = 1, 2, 3, (see (54)), being both the global and the local estimations, really near to the realerrors. It is thus apparent that the local procedure we propose is, at least in this example, a goodtool to accurately a posteriori estimate the errors in the p-version of the finite element. We canalso observe in Table III that the estimations of the global and the local errors are improvedafter adding the edge estimators. Except for the central element (in which the estimation is anupper bound for the error) the estimations remain being lower bounds of the real errors.



Table IV. Galerkin errors, local postprocessed errors and effectivityindex in the H 1(�) norm.

N ‖u − uN‖1 ‖u − (uN + Eint + Eed)‖1 �N

4 4.7279e − 2 2.2961e − 2 0.98146 1.5435e − 2 5.2776e − 3 0.99398 4.2844e − 3 1.0096e − 3 0.9812

10 9.7713e − 4 2.0000e − 4 0.9703

Table V. Galerkin errors, local postprocessed errors and effectivityindex in the L2(�) norm.

N ‖u − uN‖0 ‖u − (uN + Eint + Eed)‖0 �N

4 1.2439e − 2 4.1340e − 3 0.90486 3.0294e − 3 7.2200e − 4 0.92218 6.4026e − 4 1.1527e − 4 0.9194

10 1.1507e − 4 1.9152e − 5 0.9172

Table IV presents the global Galerkin errors in the energy norm, the global errors‖u − (uN + Eint + Eed)‖1 and the effectivity index:

�N = ‖Eint + Eed‖1

‖u − uN‖1(62)

for different values of N . All the indicators have been computed taking M = N + 2. We canobserve in Table IV that as N increases the error in the enhanced solution with M = N + 2approaches the Galerkin error corresponding to N +2, i.e. ‖u−uN+2‖1. This fact indicates thatour uncoupled procedure of adding the local interior indicators and the local edges indicatorsproduces almost the same error as uN+2, obtained as the solution of a global (coupled) discreteelliptic problem. For all the values of N in Table IV, the effectivity index �N is very close to 1.

We show in Table V, for the same values of the local degree of the polynomials as in TableIV, the error in the finite element approximation, the corrected error by adding the interiorand edge indicators and the effectivity index measured in the L2(�) norm. We can observesimilar results to those obtained in the H 1(�) norm. Again, asymptotically, the errors in theenhanced solution approach the Galerkin errors obtained setting to N + 2 the degree of thelocal polynomials. The effectivity index is also close to 1 as in the H 1(�) norm.

5.2. Elliptic problems: singular domains

We consider in this section the elliptic problem (61) in an L-shape domain

� = [−1, 1] × [−1, 1]\[0, 1] × [0, 1] (63)

with data f ≡ 1. As it is well known, the solution of this problem presents a singularity oftype r2/3.



Table VI. Galerkin errors, interior estimators and local postprocessed errors.L-shape domain. N = 6, M = 8.

‖u − uN‖1 = 2.4015e − 2‖Eint‖1 = 9.7704e − 3

‖u − (uN + Eint)‖1 = 2.1937e − 2

j = 1 j = 2 j = 3 j = 4

2.5816e − 4 2.7530e − 4 ∗ ∗i = 4 2.3720e − 4 2.4053e − 4 ∗ ∗

1.0189e − 4 1.3393e − 4 ∗ ∗5.0531e − 4 1.3962e − 2 ∗ ∗

i = 3 1.4822e − 4 3.9941e − 3 ∗ ∗4.8308e − 4 1.3379e − 2 ∗ ∗5.5938e − 4 1.3613e − 2 1.3962e − 2 2.7530e − 4

i = 2 1.6502e − 4 7.9481e − 3 3.9941e − 3 2.4053e − 45.3449e − 4 1.1052e − 2 1.3379e − 2 1.3393e − 4

2.7747e − 4 5.5938e − 4 5.0531e − 4 2.5816e − 4i = 1 2.3851e − 4 1.6502e − 4 1.4822e − 4 2.3721e − 4

1.4178e − 4 5.3449e − 4 4.8308e − 4 1.0189e − 4

In Table VI we have represented in the energy norm, the subdomain errors, the elementwise error estimations and the errors in the locally enhanced numerical approximation: thepostprocessed errors. We have used a uniform partition of the L-shape domain � into 12subdomains. Each one of the cells in the table contains the results corresponding to one ofthe subdomains of the partition while the global results (finite element error, error estimationand postprocessed error in �) are represented, as before, in the first row of the table. Thefinite element approximation has been computed using polynomials of degree N = 6 in everysubdomain of the partition. The global Galerkin error in this experiment is comparable withthe one presented in the previous section. The error estimations in each subdomain have beencomputed using only the interior basis functions with M = 8.

As in the previous numerical experiments, we can observe that the interior error estimationis always a lower bound of the corresponding subdomain error; see (54). Although the localelement-wise error estimator is able to determine the subdomains with larger error (in thisexample the elements in the neighbourhood of the singular vertex), the global error estimationis less precise than in the regular case due to the underestimation of the error in some elements.This effect can be alleviated by considering a small patch of subdomains consisting in the threeelements sharing a singular vertex. To this end, we have computed the error estimator definedas follows: let us denote by O the singular vertex. We first consider the error indicator definedby (55) in each subdomain �ij such that O /∈ �ij . We next define the patch

� = ⋃{�ij |O ∈ �ij } (64)

The error estimator in � is obtained by means of the elliptic problem, see Equation (31): findEp ∈ H 1

0 (�) such that

a(Ep, vM) = (rN , vM), ∀vM ∈ VM(�) (65)



Table VII. Galerkin errors, error estimations and postprocessed errors using apatch around the singularity. L-shape domain. N = 6, M = 8.

‖u − uN‖1 = 2.4015e − 2‖E‖1 = 1.7018e − 2

‖u − (uN + E)‖1 = 1.6944e − 2

j = 1 j = 2 j = 3 j = 4

2.5816e − 4 2.7530e − 4 ∗ ∗i = 4 2.3720e − 4 2.4053e − 4 ∗ ∗

1.0189e − 4 1.3393e − 4 ∗ ∗5.0531e − 4 1.3962e − 2 ∗ ∗

i = 3 1.4822e − 4 8.4451e − 3 ∗ ∗4.8308e − 4 9.8152e − 3 ∗ ∗5.5938e − 4 1.3613e − 2 1.3962e − 2 2.7530e − 4

i = 2 1.6502e − 4 1.2108e − 2 8.4451e − 3 2.4053e − 45.3449e − 4 9.6586e − 3 9.8152e − 3 1.3393e − 4

2.7747e − 4 5.5938e − 4 5.0531e − 4 2.5816e − 4i = 1 2.3851e − 4 1.6502e − 4 1.4822e − 4 2.3721e − 4

1.4178e − 4 5.3449e − 4 4.8308e − 4 1.0189e − 4

with M = N +2. In this experiment, N = 6 and M = 8 as in the previous one. The results arepresented in Table VII where we have denoted by E the global estimator obtained by meansof this procedure.

We can observe in Table VII that the global error estimator is much more accurate thanbefore. Furthermore, the error obtained postprocessing the Galerkin approximation with N = 6adding the locally computed error estimations is nearly the same as the error obtained with theGalerkin method with N = 8. More precisely, ‖u− (u6 +E)‖1 = 1.6944e−2 while ‖u−u8‖ =1.6916e−2. Besides, the patch-wise error estimator reproduces much more accurately the errorbehaviour in the three elements sharing the singular vertex. Obviously, the values of the errorestimators in the remaining elements are the same as those reported in Table VI. We remarkthat in practice, in problems with singular domains it is usually necessary to heavily refine themesh in the neighbourhood of the singularity; see for example, Reference [26]. It is clear thatour error estimation procedure can also be used in this case.

5.3. Non-linear evolutionary problems

We consider now a reaction–diffusion equation with a cubic non-linearity

ut = ��u + u − u3 (66)

for x = (x, y) in � = [−1, 1]× [−1, 1] subject to homogeneous Dirichlet boundary conditions.We decompose � into 16 subdomains using a uniform partition. As an initial condition, wehave taken

u(x, y, 0) = sin(/2(x + 1)) sin(/2(y + 1)) (67)



-1

0

1

2

3

-1

-0.5

0

0.5

10

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 3. Graph of the exact solution of the non-linear parabolic problem for T = 5.

We set � = 0.005 and measure the errors at times T = 1, 2, 3, 4 and 5. At the final time thesolution approaches a stable state with a relatively steep boundary layer due to the homogeneousDirichlet conditions imposed as can be seen in Figure 3. The theoretical (‘exact’) solution wascomputed with the standard Galerkin method using a number of degrees of freedom largeenough and a sufficiently small time step in the time integrator. The computed solution takenas exact is reasonably far more accurate than that shown in the experiments. For the finiteelement approximation we consider a fixed uniform mesh with 16 elements and the samedegree N for the local polynomials at each element. We denote the elements by

�i,j =[−1 + j − 1

2, −1 + j

2

]×[−1 + i − 1

2, −1 + i

2

], i, j = 1, . . . , 4

For the time integration we use a linearly implicit Runge–Kutta method of order 4 withvariable step. This method was introduced in Reference [27] where the reliability and efficiencyof this code is shown, when it is used to integrate the semidiscrete equations arising after thespatial discretization of advection–reaction–diffusion equations.

As stated in Section 2.1 we consider the elliptic problem (9) (with R = u3 − u and f = 0).We have used the local error estimation procedure of Sections 3 and 4 applied to this concreteelliptic problem. We have first computed the local interior indicator Eint and then the localedge indicator Eed. In this experiment, the computed values of Eed are so small that they donot modify the results obtained using the interior indicator alone. In consequence, they are notshown in the sequel. We have also solved the global problem (28) to get the postprocessedsolution UM

N that is known to be an improved approximation (see References [15, 16]). Thesame value of M = N +2 was selected in both local and global procedures in order to comparethe results obtained using both error estimators.



4 5 6 7 80

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Degree of the local polynomials

Err

ors

Time T=2

Galerkin Error Local Postprocess Error Global Postprocess Error

4 5 6 7 80

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08Time T=4

Err

ors

Degree of the local polynomials

Galerkin Error Local Postprocess Error Global Postprocess Error

Figure 4. Galerkin, local postprocessed and global postprocessed errors.

In Figure 4 we show the global finite element error in the H 1(�) norm, for different valuesof N , the error achieved adding the 16 local interior indicators to the Galerkin approximation(local postprocessed errors in the figure) and the postprocessed errors (global postprocessederrors in the figure). The top picture is for T = 2 and the bottom one for T = 4.



Table VIII. Galerkin errors, interior estimators and local postprocessed errors.N = 4, M = 6.

‖u − uN‖1 = 6.0135E − 2‖Eint‖1 = 5.7901e − 2

‖u − (uN + Eint)‖1 = 2.8845E − 2

j = 1 j = 2 j = 3 j = 4

1.2899e − 2 1.9202e − 2 1.9202e − 2 1.2899e − 2i = 4 1.3959e − 2 1.7934e − 2 1.7934e − 2 1.3959e − 2

6.8363e − 3 8.9746e − 3 8.9746e − 3 6.8363e − 3

1.9202e − 2 4.4222e − 4 4.4222e − 4 1.9202e − 2i = 3 1.7934e − 2 6.8856e − 5 6.8850e − 5 1.7934e − 2

8.9747e − 3 4.3060e − 4 4.3059e − 4 8.9747e − 3

1.9202e − 2 4.4224e − 4 4.4223e − 4 1.9202e − 2i = 2 1.7934e − 2 6.8871e − 5 6.8863e − 5 1.7934e − 2

8.9747e − 3 4.3061e − 4 4.3060e − 4 8.9747e − 3

1.2899e − 2 1.9202e − 2 1.9202e − 2 1.2899e − 2i = 1 1.3959e − 2 1.7934e − 2 1.7934e − 2 1.3959e − 2

6.8363e − 3 8.9746e − 3 8.9746e − 3 6.8363e − 3

The figure shows the exponential convergence that is expected when a smooth solutionis approximated by means of the p-version of the finite element method; see for example,Reference [26]. We observe that both local and global procedures reduce the global Galerkinerror. A bigger reduction of the error is observed for the postprocessed Galerkin method (theglobal procedure) although for most of the values shown in the figure, the local procedureachieves a similar reduction of the error than that corresponding to the postprocessed method.Besides, we remark that, in this experiment, the error is strongly reduced both when the solutionis far from the stationary state (T = 2) or close (T = 4) to the stationary state. Furthermore, wecan also observe that both local and global procedures work even if the asymptotic convergenceregime is not reached yet, see the errors reported in the bottom picture (T = 4) for N = 4.

In Table VIII we show for time T = 2 and N = 4 the H 1(�) norm of the error in thefinite element approximation, the interior error estimator and the local postprocessed error, bothmeasured in the H 1(�) norm. The first row is for the global errors while the rest of the datacorrespond to the local Galerkin errors, local interior estimators and local postprocessed errorsat each of the 16 subdomains �i,j , i, j = 1, . . . , 4.

We can observe that the errors at the four central subdomains are significantly smaller thanthe others and that

• the local interior estimators detect the four central subdomains that have a much smallerlocal Galerkin error

• the local real errors are very accurately estimated using the local interior estimators inthe 12 surrounding subdomains with dominant error

• the errors in all the subdomains are improved after adding the interior estimators. Theimprovement is significant in those subdomains with bigger error.



-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-0.02

-0.01

0

0.01

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1-0.02

-0.01

0

0.01

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-0.02

-0.01

0

0.01

Figure 5. Galerkin error function, function Eint and function uN −UMN at time T = 2. N = 4, M = 6.

To study in some detail the behaviour of the error estimators, in Figure 5 we have shownfor the same values of Table VIII (N = 4 and T = 2) the error function in the Galerkinapproximation u(·, 2)−uN(·, 2) (top picture), the error estimator function for the local procedureEint(·) (left bottom picture) and the error estimator function for the global postprocess of theGalerkin approximation uN(·, 2)−UM

N (·, 2) (right bottom picture). We can observe that the localprocedure not only catches the size of the errors in each subdomain but also the qualitativebehaviour of the errors. In fact, comparing the behaviour of Eint(·) and uN(·, 2) − UM

N (·, 2),it is quite clear that the difference is due exclusively to the fact that the inter-element errorsare not represented in the local estimator Eint(·) as a consequence of the use of bubble-typefunctions. As has been indicated before, in this experiment adding edge error indicators as inSection 4 does not add any noticeable change in the estimation. Whether other possible localprocedures as the use of Neumann-type boundary conditions can improve the procedure is thesubject of further research.



Table IX. Effectivity indices for T = 2 and 4.

T = 2 T = 4

N �1N �2

N �1N �2

N

4 0.9654 1.0389 0.8929 0.96025 1.1686 1.2373 1.0227 1.08266 1.2522 1.3243 1.0714 1.12747 1.0840 1.1365 1.0645 1.11628 1.0155 1.0572 0.8541 0.8894

Finally, in Table IX we show the effectivity indices for two different values of the final timeT (T = 2 and 4). The first index is obtained with the interior estimator and the second withthe (global) postprocessed solution.

We use the following notation:

�1N = ‖Eint‖1

‖u − uN‖1, �2

N = ‖uN − UNM‖1

‖u − uN‖1(68)

From Table IX we deduce that:

• The local interior estimations are always smaller than the postprocessed estimations. Thishas been proven in the paper and can be checked in the tables. The values of �1

N aresmaller than �2

N .• For some values the local interior estimators are lower bounds of the real errors while for

others they are upper bounds of the real errors. This is different from the elliptic case inwhich the interior estimator gives always a lower bound.

• The local procedure proposed to estimate the errors works really well in this non-linearevolutionary example giving even more values of the effectivity index than those corre-sponding to the global procedure for most of the data of Table IX.

ACKNOWLEDGEMENTS

This research has been supported by DGI-MCYT under project BFM2001-2138 (cofinanced by FEDERfunds) and by JCYL under project VA044/03.

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Element-wise a posteriori estimates based on hierarchical bases for non-linear parabolic problems

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