A posteriori error analysis for stabilised finite element approximations of transport problems

A posteriori error analysis for stabilised ®nite elementapproximations of transport problems

Paul Houston a, Rolf Rannacher b, Endre S�uli a,*

a Oxford University, Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UKb Institut f�ur Angewandte Mathematik, Universit�at Heidelberg, Im Neuenheimer Feld 293, D-69120 Heidelberg, Germany

Received 5 May 1999

Abstract

We develop the a posteriori error analysis of stabilised ®nite element approximations to linear transport problems via duality

arguments. Two alternative dual problems are considered: one is based on the formal adjoint of the hyperbolic di�erential operator, the

other on the transposition of the bilinear form for the stabilised ®nite element method. We show both analytically and through nu-

merical experiments that the second approach is superior in the sense that it leads to sharper a posteriori error bounds and more

economical adaptively re®ned meshes. Ó 2000 Elsevier Science S.A. All rights reserved.

MSC: 65N12; 65N15; 65N30

Keywords: Stabilised ®nite element methods; a posteriori error analysis; Hyperbolic problems

1. Introduction

The purpose of this article is to develop the a posteriori error analysis of stabilised ®nite element ap-proximations to transport problems, including the streamline di�usion ®nite element method and the least-squares stabilised ®nite element method (see [9]). For simplicity, we restrict the discussion to the case of ascalar linear transport equation of the form

b � ru� cu � f �1:1�

subject to appropriate in¯ow boundary condition. Generalisations to systems and nonlinear equations arepossible.

Mesh adaptation in ®nite element discretisations should be based on rigorous a posteriori error esti-mates; for hyperbolic equations such estimates should re¯ect the inherent mechanisms of error propagation(see [6,8]). These considerations are particularly important when local quantities such as point values, localaverages or ¯ux integrals of the analytical solution are to be computed with high accuracy. Selective errorestimates of this kind can be obtained by the optimal control technique proposed in [2,5] which is based onduality arguments analogous to those from the a priori error analysis of ®nite element methods. In theresulting a posteriori error estimates the element-residuals of the computed solution are multiplied by localweights involving the dual solution. These weights represent the sensitivity of the relevant error quantitywith respect to variations of the local mesh size. Since the dual solution is usually unknown analytically, ithas to be approximated numerically. On the basis of the resulting a posteriori error estimate the current

www.elsevier.com/locate/cmaComput. Methods Appl. Mech. Engrg. 190 (2000) 1483±1508

* Corresponding author.

0045-7825/00/$ - see front matter Ó 2000 Elsevier Science S.A. All rights reserved.

PII: S 0 0 4 5 - 7 8 2 5 ( 0 0 ) 0 0 1 7 4 - 2

mesh is locally adapted and then new approximations to the primal and the dual solution are computed.This feed-back process is repeated, for instance, until the required error tolerance is reached. In this way,optimal meshes can be obtained for various kinds of error measures, where optimal can mean most eco-nomical for achieving a prescribed accuracy TOL or most accurate for a given maximum number Nmax of meshcells. This approach is quite universal as it can, in principle, be applied to almost any problem, linear ornonlinear, as long as it is posed in a variational setting. For a collection of examples, we refer to the surveyarticle [13] (see also [3,12]).

A new feature of this technique, when applied to stabilised ®nite element methods, is the choice of thedual problem, and it is the study of this question that represents the subject of the present paper. Inparticular, we show that the naive approach of using the natural dual based on the formal adjoint of thepartial di�erential operator results in a posteriori error estimates in which the stabilisation terms maystrongly dominate the other residual terms, leading to over-re®nement of the mesh. We then show that thisde®ciency can be overcome by exploiting the dual of the `stabilised' primal di�erential operator. In this way,extra powers of the mesh size are gained in the stabilisation terms, resulting in an optimal-order errorbound. The success of the latter approach is explained by theoretical analysis and is illustrated in detail bynumerical experiments.

The outline of the paper is as follows. We start, in Section 2, by describing the key ideas on a one-dimensional model problem. In Section 3, guided by the one-dimensional analysis, we then consider thequestion of a posteriori error estimation for stabilised Galerkin approximations to multi-dimensionaltransport equations. In particular, we discuss the problem of a posteriori error estimation for norms as wellas linear functionals of the solution. Section 4 presents a series of numerical experiments which illustrateand verify the theoretical results. Finally, in Section 5 we summarise the work presented in this paper anddraw some conclusions.

2. The analysis of the one-dimensional problem

In order to highlight the key issues concerning the error analysis while avoiding unnecessary techni-calities, we begin by considering the following model problem:

u0�x� � f �x� for x 2 �0; 1�; u�0� � g; �2:1�where f 2 L2�0; 1� and g is a given real number; it is a simple matter to show that this has a unique solutionin H 1�0; 1�.

Now suppose that the interval �0; 1� has been partitioned by a nonuniform mesh de®ned by the mesh-

points fxigNi�0, where 0 � x0 < x1 < � � � < xNÿ1 < xN � 1; and N is a positive integer, N P 1. We let

Ij � �xjÿ1; xj�, hj � xj ÿ xjÿ1, and introduce the mesh function h de®ned on �0; 1� by h�x� � hj for x 2 Ij,j � 1; . . . ;N . On this partition we consider the ®nite element space Sh, Sh � H 1�0; 1�, consisting of con-tinuous piecewise polynomials of ®xed degree p, p P 1. We recall the following standard approximationproperty of the ®nite element interpolant Ph : H 1�0; 1� ! Sh (see [4]):

(A) Given that v 2 Hs�1�0; 1� for some s, 06 s6 p, there exists a constant Ci, independent of v and h, suchthat

kvÿ PhvkL2�Ij� � hjjvÿ PhvjH1�Ij�6Cihs�1j jvjHs�1�Ij� �2:2�

for all j � 1; . . . ;N . Here, j � jHr�Ij� denotes the usual seminorm of the Sobolev space H r�Ij�, r P 1.Next, we introduce the streamline di�usion ®nite element approximation of (2.1); to do so, we consider a

positive function d 2 L1�0; 1� called the streamline diffusion parameter, the bilinear form Bd��; �� and thelinear functional ld de®ned by

Bd�w; v� � �w0; v� dv0� � w�0�v�0�; ld�v� � �f ; v� dv0� � gv�0�;respectively. In these de®nitions ��; �� denotes the inner product of L2�0; 1�.

The streamline diffusion method is: ®nd uh 2 Sh such that

Bd�uh; vh� � ld�vh� 8vh 2 Sh: �2:3�

1484 P. Houston et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 1483±1508

As Bd�vh; vh� > 0 for all vh in Sh n f0g and Sh is a ®nite-dimensional linear space, it follows that (2.3) has aunique solution uh in Sh. Formally, (2.3) can be viewed as a perturbation of the standard Galerkin methodcorresponding to d � 0.

Here we shall be concerned with the a posteriori error analysis of the streamline di�usion method: ouraim is to derive a bound on the global error e � uÿ uh in terms of the computable ®nite element residuals

rh � f ÿ u0h; rÿh � g ÿ uh�0�;which arise from inserting uh into the di�erential equation in (2.1) and the associated boundary condition atin¯ow. The analysis relies on the following Galerkin property:

Bd�uÿ uh; vh� � 0 8vh 2 Sh: �2:4�The identity (2.4) is easily seen to hold by noting (2.3) and that

Bd�u; vh� � ld�vh� 8vh 2 Sh:

In Sections 2.1 and 2.2 we shall describe two distinct approaches to the a posteriori error analysis of thestreamline di�usion method; being driven by duality arguments, they both proceed in the same manner, butsince they use di�erent dual problems the resulting a posteriori error bounds are not identical.

2.1. The ®rst approach and its limitations

The starting point in the argument is the following dual problem: given that w 2 C1�0; 1�, ®nd z such that

ÿz0 � w�x�; x 2 �0; 1�; z�1� � 0; �2:5�clearly, this has a unique solution z 2 C1�0; 1�. Now we are ready to state our ®rst result.

Lemma 1. The following bound holds for 06 s6 p:

j�uÿ uh;w�j6Ci

XN

j�1

khrhkL2�Ij� min06 r6 s

hrj jzjHr�1�Ij�

n o� �1� Ci�

XN

j�1

kdrhkL2�Ij�jzjH1�Ij�; �2:6�

where Ci is the constant from (2.2).

Proof. Recalling the dual problem (2.5), integrating by parts, and appealing to the Galerkin property (2.4),we deduce that, for any zh 2 Sh,

�uÿ uh;w� � �rh; zÿ zh� ÿ �drh; z0h� � rÿh �z�0� ÿ zh�0�� I� II� III: �2:7�Next, given that s is a ®xed real number, 06 s6 p, we take zh � Phz, where Ph is the ®nite element inter-polant of v from Sh. Then, z�0� ÿ zh�0� � z�0� ÿ �Phz��0� � 0, yielding III � 0. Let us ®rst deal with term I.Noting (2.2), we conclude

jIj6Ci

XN

j�1

khrhkL2�Ij� min06r6 s

fhrj jzjHr�1�Ij�g:

For term II, on writing Phz � z� �Phzÿ z� and applying (2.2) we have

j II j6XN

j�1

kdrhkL2�Ij�k�Phz�0kL2�Ij�6 �1� Ci�XN

j�1

kdrhkL2�Ij�jzjH1�Ij�:

Collecting the bounds on I and II and inserting them into (2.7), we deduce (2.6). �

In order to ensure that the ®rst term in this a posteriori error bound fully re¯ects the approximationproperties of the test space, one would wish to choose s as large as possible, namely s � p. However, since in

P. Houston et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 1483±1508 1485

the second term rh is multiplied by d rather than hs�1, the error bound (2.6) will be properly balanced only ifd6Cdhs�1 where Cd is a ®xed positive constant. For s > 0, such choice of d is considerably smaller than thestandard one of d � Cdh (cf. [9,10]), and may lead to an under-stabilised numerical approximation whichexhibits nonphysical numerical oscillations. Thus, from the practical point of view, there appears to be nobene®t from choosing any value of s other than the suboptimal one of s � 0; indeed, with s � 0 in (2.6) andnoting that jzjH1�0;1� � kwkL2�0;1�; we immediately deduce the next result.

Corollary 2. The following a posteriori error bound holds:

kuÿ uhkL2�0;1�6CikhrhkL2�0;1� � �1� Ci�kdrhkL2�0;1�:

In the next section, we present a strategy which overcomes the undesirable features of this ®rst approach.

2.2. The second approach

Although (2.5) is the most natural candidate for the dual problem, it is by no means the only possiblechoice. Indeed, as will be seen below, the a posteriori error bound (2.6) can be improved by selecting a dualproblem which respects the particular structure of the Galerkin property (2.4).

Thus, we now consider the following dual problem: given that w is in C1�0; 1�, ®nd zd in H 1�0; 1� suchthat

Bd�w; zd� � �w;w� 8w 2 H 1�0; 1�: �2:8�Here, the solution to the dual problem is denoted by zd in order to emphasise the dependence on thestabilisation parameter d. Recalling the de®nition of Bd, a simple calculation based on integration by partsshows that (2.8) can be restated as follows:

ÿ �zd � dz0d�0 � w�x�; x 2[Nj�1

�xjÿ1; xj�;

�zd � dz0d��xj� � 0; j � 1; . . . ;N ÿ 1;

z0d�0� � 0; �zd � dz0d��1� � 0;

�2:9�

where �w��x� � w�x�� ÿ w�xÿ� denotes the jump of w at x. In particular, when d 2 W 11�0; 1�, the function

zd � dz0d is continuous on �0; 1� and the jump conditions are trivially satis®ed. In this case, the dual problemcollapses to

ÿ �zd � dz0d�0 � w�x�; x 2 �0; 1�;z0d�0� � 0; �zd � dz0d��1� � 0

�2:10�

To motivate the hypotheses which we shall make on w, we give the following characterisation of the dualsolution.

Lemma 3. Suppose that d is a constant, d > 0, and let n be a nonnegative integer; then

zd�x� �Z 1

xw�n�dn�

Xn

m�1

�ÿ1�mÿ1dm�w�mÿ1��x� ÿ eÿx=dw�mÿ1��0�� ÿ1�ndneÿx=d

Z x

0

w�n��n�en=d dn:

�2:11�When n � 0 the index set of the summation in the second term on the right is empty, and the corresponding sumis then defined to be identically zero.

Proof. Let n � 0; integrating the di�erential equation in (2.10) between x and 1 we deduce, recalling theboundary condition at x � 1, that


zd � dz0d �Z 1

xw�n�dn; x 2 �0; 1�: �2:12�

Solving the di�erential equation (2.12) gives

zd�x�ex=d ÿ zd�0� �Z x

0

v�n��en=d�0 dn; where v�x� �Z 1

xw�n�dn:

Integrating by parts on the right and noting the boundary condition at x � 0 in (2.10), which, according to(2.12), is equivalent to demanding that zd�0� � v�0�, we deduce that

zd�x� � v�x� � eÿx=d

Z x

0

w�n�en=d dn;

which is just the required identity for n � 0. For n P 1, we write under the integral sign on the right in thelast equality

en=d � dn dn

dnn en=d;

and integrate by parts n times to deduce (2.11). �

It is clear from this lemma that derivatives of zd will exhibit a boundary layer at x � 0, unless the functionw and its derivatives vanish at this point. Armed by this observation, we state the following result.

Lemma 4. (a) Let d 2 W 11�0; 1� and suppose that there exists a constant K1 such that

1� 1

2d0�x�P K1 > 0 for a:e: x 2 �0; 1�;

then, there is a positive constant M1 � M1�K1� such that, for each w 2 C1�0; 1�,jzdjH1�0;1�6M1kwkL2�0;1�: �2:13�

(b) Let 06m6 p ÿ 1, d 2 W m�21 �0; 1� and suppose that there exist constants Lm�2 and Km�2 such that

jd�m�2��x�j6 Lm�2 for a.e. x 2 �0; 1� and

1� m�� 3

2

�d0�x�P Km�2 > 0 for a:e: x 2 �0; 1�;

then, there exists a positive constant Mm�2 � Mm�2�Km�2;M1; . . . ;Mm�1; L2; . . . ; Lm�2� such that for eachw 2 C1�0; 1� with w�l��0� � 0 for all l � 0; . . . ;m, we have

jzdjHm�2�0;1�6Mm�2jwjHm�1�0;1�: �2:14�

Proof. (a) To prove (2.13), we di�erentiate (2.12) to deduce that

�1� d0�z0d � dz00d � ÿw�x�; x 2 �0; 1�: �2:15�Taking the L2�0; 1� inner product of (2.15) with z0d and noting that z0d�0� � 0, upon manipulating the secondterm on the left and applying the Cauchy±Schwarz inequality on the right, we arrive at

K1kz0dk2L2�0;1� �

1

2d�1�jz0d�1�j26 kwkL2�0;1�kz0dkL2�0;1�:

Hence (2.13) with M1 � 1=K1.(b) For m P 0 the proof is analogous, by induction: on di�erentiating (2.12) m� 2 times and taking the

L2�0; 1� inner product of the resulting equality with z�m�2�, after a simple manipulation based on integrationby parts we obtain the inequality (2.14) where


Mm�2 � Nm�2

Km�2

1

"�Xm�2

k�2

m� 2k

� �LkMm�3ÿk

#;

and Nm�2 is the constant in the following Poincar�e inequality:

kwkHm�1�0;1�6Nm�2jwjHm�1�0;1�: �

Now we are ready to embark on the a posteriori error analysis of the streamline di�usion ®nite elementmethod, using the dual problem (2.8). We state the following analogue of Lemma 1.

Lemma 5. Suppose that 06 s6 p and let d 2 W s�11 �0; 1�; then,

j�uÿ uh;w�j6Ci

XN

j�1

khrhkL2�Ij��

� kdrhkL2�Ij��

min06r6 s

fhrj jzdjHr�1�Ij�g; �2:16�

where Ci is the constant from (2.2).

Proof. By the de®nition of the dual problem (2.8), with w � uÿ uh, and the Galerkin property (2.4), wehave that

�uÿ uh;w� � Bd�uÿ uh; zd� � Bd�uÿ uh; zd ÿ zd;h�� rh; zd ÿ zd;h� � �rh; d�z0d ÿ z0d;h�� rÿh �zd�0� ÿ zd;h�0��:

For s ®xed, 06 s6 p, we choose zd;h � Phzd and proceed in the same way as in the proof of Lemma 1, toarrive at the desired bound. �

We note in passing that when d is a constant function on �0; 1� the hypotheses of Lemma 4 are satis®edwith K1 � K2 � � � � � Km�2 � 1. Tracing the constants in the proof, we then ®nd thatM1 � M2 � � � � � Mm�2 � 1. In general, d will not be constant on �0; 1�, so the assumptions on d stated in a)and b) of Lemma 4 can be seen as conditions on the variation of the computational mesh, given that inpractice d is related to h. Let us suppose, for example, that d is de®ned as a piecewise linear function whosevalue at the midpoint of Ij is equal to hj, the length of the interval Ij. The hypothesis on d from part a) isthen satis®ed on any mesh which results from a coarse background mesh through successive local bisec-tions, so it does not represent a practical limitation from the point of view of our adaptive mesh re®nementalgorithm. The conditions which occur in part b) are increasingly more demanding; for large values of mthey require that the variation (more precisely, the compression from left to right) of the mesh is small.However, we expect that the mesh-size distribution and hence d will approach a piecewise smooth functionin the process of mesh adaptation. The potential loss of regularity in d and the consequential loss ofsmoothness in zd in some interval Ij is compensated by the presence of the factor hr

j in (2.16). Indeed, theestimate (2.16) is at least as sharp as the one of Lemma 1. We shall return to this question again in Section 3in the error analysis of the multi-dimensional problem.

The a posteriori error bound resulting from Lemma 5 is stated below in Corollary 6. First we introducesome notation. For k a positive integer, we put

_Hk�0; 1� � fv 2 H k�0; 1� : v�0� � � � � � v�kÿ1��0� � 0g:

For k � 0, by de®nition, _H 0�0; 1� :� L2�0; 1�. Further, we introduce the negative Sobolev norm k � kHÿk�0;1�by

kwkHÿk�0;1� � supv2 _Hk�0;1�

j�w; v�jjvjHk�0;1�

:


In particular, kwkH0�0;1� � kwkL2�0;1�. On noting that C1�0; 1� is dense in L2�0; 1� � _H 0�0; 1� and that the setfw 2 C1�0; 1� : w�0� � . . . � w�kÿ1��0� � 0g is dense in _Hk�0; 1�, k P 1, we deduce from Lemmas 4 and 5 thefollowing a posteriori error bound for the streamline di�usion method.

Corollary 6. Suppose that 06 s6 p and let the conditions of Lemma 4 hold with m � sÿ 1 (when s � 0, theconditions in part a) of Lemma 4 are assumed); then the following a posteriori error bound holds:

kuÿ uhkHÿs�0;1�6CiMs�1 khs�1rhkL2�0;1��

� khsdrhkL2�0;1��:

Unlike the a posteriori error bound discussed in the previous section, this estimate now re¯ects com-pletely the approximation properties of the test space, with the practically realistic value of d � Cdh.

3. The multi-dimensional model problem

We consider the hyperbolic boundary-value problem

Lu � b � ru� cu � f in X; u � g on Cÿ; �3:1�

where X � �0; 1�n, with inflow boundary Cÿ � fx 2 oX : b�x� � m�x� < 0g; here m�x� denotes the unit outwardnormal vector at x 2 oX (de®ned almost everywhere on oX). Analogously, we de®ne the outflow boundaryC� � fx 2 oX : b�x� � m�x�P 0g.

We shall suppose that the entries b1; . . . ; bn of the n-component vector function b are continuouslydifferentiable and positive on �X; this ensures that oX is noncharacteristic for L at almost every point x onoX. It will be assumed that c 2 C��X�, f 2 L2�X� and g 2 L2�Cÿ�.

In order to set up the variational formulation of (3.1), we associate with L the function space

H�X� � fv 2 L2�X� : Lv 2 L2�X�g;

we consider the bilinear form B��; �� : H�X� � H�X� ! R de®ned by

B�w; v� � �Lw; v� ÿ ��b � m�w; v�Cÿand introduce the linear form l : H�X� ! R by

l�v� � �f ; v� ÿ ��b � m�g; v�Cÿ :In these de®nitions ��; �� denotes the L2 inner product over X and ��; ��Cÿ signi®es the L2 inner product overCÿ (with ��; ��C� being de®ned analogously). It can be shown that any function v 2 H�X� has a well de®nedtrace on Cÿ (respectively, C�) in L2�Cÿ� (respectively, L2�C��), so the de®nitions of B��; �� and l�� aremeaningful. When equipped with the graph-norm

kvkH�X� � �kvk2L2�X� � kLvk2

L2�X��1=2;

H�X� is a Hilbert space. With this notation, the boundary-value problem (3.1) can be expressed as follows:®nd u in H�X� such that

B�u; v� � l�v� 8v 2 H�X�: �3:2�It is a straightforward matter to prove that under the present hypotheses on b, c, f and g problem (3.2) has aunique solution u in H�X�.

Let Th � fjg be a ®nite element partition of X into open element domains j. We shall suppose that thefamily of partitions fThg is shape regular; namely, there exists a positive constant a such that

hj

qj

6 a 8j 2[

h

Th;


with hj � diam��j� and qj the diameter of the largest ball contained in �j. Let h�x� denote the mesh functionwhose value on element j is equal to hj. On Th we consider the ®nite element space Sh � H 1�X� �� H�X��containing continuous piecewise polynomials of maximum degree p, p P 1. It will be assumed that Sh

possesses the following standard approximation property:(B) Given that v 2 Hs�1�X� and vjCÿ 2 H s�1�Cÿ� for some s, 06 s6 p, there exists vh in Sh and a constant

Ci, independent of v and the mesh function h, such that

kvÿ vhkL2�j� � hjjvÿ vhjH1�j�6Cihs�1j jvjHs�1�j�; �3:3�

kvÿ vhkL2�oj\Cÿ�6Cihs�1j jvjHs�1�oj\Cÿ�; �3:4�

for all j 2Th. Here and in the rest of the paper, for s6 n=2, j signi®es the union of all such elements whoseclosure has nonempty intersection with the closure of j; for s > n=2, we de®ne j � j. Given a function vand an associated function vh satisfying (3.3), (3.4), we shall write vh � Phv to denote that vh is assigned to v.

Remark 7. For s > n=2 hypothesis (B) can be satis®ed by selecting Phv as the ®nite element interpolant ofv 2 H s�1�X� from Sh. For s6 n=2, point evaluation of v need not be meaningful, due to the lack of Sobolevregularity; in this case Phv can be chosen as a suitable quasi-interpolant of v. A further possibility is to de®nePhv 2 Sh at degrees of freedom interior to X [ C� as indicated in the previous sentence, while on Cÿ one cande®ne PhvjCÿ as the orthogonal projection of vjCÿ onto the restriction of Sh to Cÿ with respect to the innerproductZ

Cÿjb � mjw�x�v�x�ds:

This latter choice of Ph will be made use of in Remark 17 below.

Next, we introduce the ®nite element approximation of (3.2); to do so, we consider the stabilisationparameter as a positive function d contained in L1�X�, the bilinear form Bd��; �� de®ned on H�X� � H�X� by

Bd�w; v� � �Lw; v� dLv� ÿ ��b � m�w; v�Cÿ ; �3:5�

where following Baiocchi and Brezzi [1], we put Lw � b � rw� cw with

c �0 for the streamline diffusion finite element method;c for the least-squares stabilised finite element method;r � bÿ c for the negative-adjoint stabilised finite element method;

8<:and introduce the linear functional ld�� : H�X� ! R by

ld�v� � �f ; v� dLv� ÿ ��b � m�g; v�Cÿ :

With these notations, we formulate the stabilised Galerkin finite element method: ®nd uh 2 Sh such that

Bd�uh; vh� � ld�vh� 8vh 2 Sh: �3:6�

Remark 8. In the case of the streamline di�usion method and the least-square stabilised ®nite elementmethod, assuming, for example, that there exists a nonnegative constant c0 such that

cÿ 1

2r � b P c0 on �X; �3:7�

it is a straightforward matter to prove that Bd�vh; vh� > 0 for all vh in Sh n f0g. The same is true in the case ofthe negative-adjoint stabilised ®nite element method, provided that (3.7) holds with c0 > 0 andd6 1

2c0�c2 � �r � b�2�ÿ1

on �X. As Sh is ®nite-dimensional, it follows for each of the three choices of L abovethat (3.6) has a unique solution uh in Sh.


3.1. A posteriori error analysis for norms

The a posteriori error analysis of (3.6) is based on considering the ®nite element residual and theboundary residual de®ned, respectively, by

rh � f ÿLuh; rÿh � �g ÿ uh�jCÿ ;and noting the Galerkin property

Bd�uÿ uh; vh� � 0 8vh 2 Sh: �3:8�As in the second approach in one-dimension, we proceed by introducing the stabilisation-dependent dualproblem: ®nd zd 2 H�X� such that

Bd�w; zd� � �w;w� 8w 2 H�X�; �3:9�where w is a given function in C10 �X�. The existence of a unique solution zd in H�X� to (3.9) follows byapplying the Lax±Milgram Theorem; see also Theorem 1.4.1 in the monograph of Oleinik and Radkevi�c[11]. For the moment we shall suppose that zd is suf®ciently smooth, namely zd 2 H s�1�X�, 06 s6 p; con-ditions which ensure this will be stated below in Lemma 10. We begin with the following preliminary result.

Lemma 9. Suppose that 06 s6 p and zd 2 H s�1�X�; then,

j�uÿ uh;w�j6CiN1

Xe : e\Cÿ6�;

kh1=2e rÿh kL2�e� min

06r6 sfhr

e jzdjHr�1�je�g

� Ci

Xj2Th

khjrhkL2�j��

� N2kdrhkL2�j��

min06r6 s

fhrjjzdjHr�1�j�g;

�3:10�

where N1 ��2C3

p kbkL1�Cÿ�, N2 � kbkL1�X� � khckL1�X�, Ci is the constant from condition (B), C3 is a positiveconstant that depends only on the mesh-regularity parameter a and the number of space dimensions n, je is theelement with (open) face e and he is the diameter of e.

Proof. Choosing w � uÿ uh in (3.9) and noting the Galerkin property, we deduce that

�uÿ uh;w� � Bd�uÿ uh; zd� � Bd�uÿ uh; zd ÿ zd;h�� rh; zd ÿ zd;h� � �rh; dL�zd ÿ zd;h�� ÿ ��b � m�rÿh ; �zd ÿ zd;h��Cÿ� I� II� III; �3:11�

where zd;h � Phzd (cf. (B) above). We shall proceed by estimating each of the terms I, II and III by means ofthe approximation property (B). First, we have

jIj6Ci

Xj2Th

khjrhkL2�j� min06 r6 s

fhrjjzdjHr�1�j�g:

Second,

jIIj6Ci

Xj2Th

�kbkL1�j� � hjkckL1�j��kdrhkL2�j� min06r6 s

fhrjjzdjHr�1�j�g:

Third, denoting by e an open face of an element in the partition Th, we deduce that

jIIIj6X

e : e\Cÿ6�;kbkL1�e�krÿh kL2�e�kzd ÿ zd;hkL2�e�:

To each open face e which has nonempty intersection with Cÿ we assign the element je in Th with that face.Now,

kzd ÿ zd;hk2L2�e�6C3 kzd

�ÿ zd;hkL2�je�jzd ÿ zd;hjH1�je� �

1

hje

kzd ÿ zd;hk2L2�je�

�;


where C3 is a positive constant which only depends on the mesh-regularity parameter a and the number ofspace dimensions, n. Consequently, by (B),

kzd ÿ zd;hkL2�e�6Ci

��2C3

phs�1=2

jejzdjHs�1�je�:

Inserting this into the bound on term III yields, possibly with an altered value of C3 � C3�a; n�,

jIIIj6Ci

��2C3

p Xe : e\Cÿ6�;

kbkL1�e�kh1=2e rÿh kL2�e� min

06r6 sfhr

e jzdjHr�1�je�g:

Upon substituting the bounds on I, II and III into (3.11), we obtain (3.10). �

Now we shall verify that zd 2 Hs�1�X� and derive a bound on jzdjHs�1�X� in terms of the H s�1�X� norm of w.

Lemma 10. Suppose that s P 0, b 2 �Cs�2��X��n, c 2 Cs�1��X�, d 2 W s�11 �X�, w 2 C10 �X�, and assume that

there exists positive constants Ls�1 and Ks�1 such that

maxjaj6 s�1

jDad�x�j6 Ls�1; 1ÿ �s� 1�jrd�x�j1jb�x�j1 P Ks�1 8x 2 X; �3:12�

where j � j1 denotes the l1 vector norm on Rn. Let us also assume that r � bÿ cÿ c � 0 in a small but fixedneighbourhood of the origin, independent of d; then, zd 2 H s�1�X� and there exists a constantMs�1 � Ms�1�Ks�1; Ls�1�, independent of d and w, such that

kzdkHs�1�X�6Ms�1kwkHs�1�X�: �3:13�

We note here that r � bÿ cÿ c � 0 is a compatibility condition between the data of the partial di�erentialequation and the homogeneous in¯ow boundary datum at the vertex point of the in¯ow boundary.

Proof. Since d 2 W 11�X�, (3.9) can be rewritten in the following strong form:

ÿr � �b�zd � d�b � rzd � czd�� c�zd � d�b � rzd � czd�� w in X;

b � rzd � czd � 0 on Cÿ; �3:14�zd � d�b � rzd � czd� � 0 on C�:

Putting v � zd � d�b � rzd � czd�, we can write (3.14) as a system of coupled scalar hyperbolic equations:

ÿr � �bv� � cv � w in X; v � 0 on C�; �3:15�zd � d�b � rzd � czd� � v on X; zd � v on Cÿ: �3:16�

First we shall bound v in terms of w, then zd in terms of v; combining the two will yield the desired bound onzd in terms of w.

Recalling that w 2 C10 �X�, it follows by the Differentiability Theorem of Rauch ([14, p. 272]) applied to(3.15) that

kvkHs�1�X�6Ms�1kwkHs�1�X�; �3:17�where Ms�1 is a positive constant, independent of w (and of d, of course). Next we shall derive bounds onSobolev norms of zd in terms of v, independent of the stabilisation parameter d. We begin by reformulating(3.16) so that the boundary condition is homogeneous: we put w � zd ÿ v and observe that w is the solutionof the following boundary value problem

b � rw� c�� 1

d

�w � ÿ�b � rv� cv� in X; w � 0 on Cÿ: �3:18�


Further, denoting the forcing function in (3.18) by /, we see from (3.15) that

/ � ÿ�b � rv� cv� � w� v�r � bÿ cÿ c�: �3:19�With this new notation (3.18) becomes

db � rw� �1� dc�w � d/ in X; w � 0 on Cÿ:

Since w 2 C10 �X� and r � bÿ cÿ c vanishes in the vicinity of the origin, it follows that / 2 H s�1�X� and /vanishes in the vicinity of the origin. Thus, from Rauch's theorem, we deduce that w 2 Hs�1�X� and

kwkHs�1�X�6Ms�1k/kHs�1�X�; �3:20�where Ms�1 is a positive constant (possibly di�erent than in (3.17)). The independence of Ms�1 on d followsby tracing the constants in Rauch's proof, making use of (3.12). Finally, we recall that zd � v� w and applythe bounds on v and w. �

Now we are ready to complete the a posteriori error analysis we embarked on before the statement ofLemma 10. In particular, we derive a posteriori bounds on negative Sobolev norms of the global erroruÿ uh. Given that m is a positive integer, we de®ne H m

0 �X� as the closure of C10 �X� in the norm of H m�X�and introduce the negative Sobolev norm k � kHÿm�X� in the usual way by

kwkHÿm�X� � supv2C1

0�X�

j�w; v�jkvkHm�X�

:

The next result follows by applying (3.13) to the right-hand side of (3.10).

Theorem 11. Suppose that 06 s6 p and assume that the conditions of Lemma 10 hold; then, we have thefollowing a posteriori error bound:

kuÿ uhkHÿsÿ1�X�6Ms�1 Ci

Xj2Th

khs�1j rhk2

L2�j�

8<: � N 22 khs

jdrhk2L2�j�

!1=2

� CiN1

Xe : e\Cÿ6�;

khs�1=2e rÿh k2

L2�e�

!1=29=;;

where Ms�1 is a positive constant, independent of d, Ci is the constant from condition (B), and N1 and N2 are asin Lemma 9.

3.2. A posteriori error estimation for linear functionals

In many problems of physical importance the quantity of interest is a linear functional of the solution.Relevant examples include the lift and drag coe�cients for a body immersed into an inviscid ¯uid, the localmean value of the ®eld, or its ¯ux through the out¯ow boundary of the computational domain.

Suppose that we wish to control the discretisation error in some linear functional J�� de®ned on thespace H�X� (or on a suitable subspace which contains the ®nite element space Sh and the exact solution u ofproblem (3.1)). To do so, following the same line of thought as in the one-dimensional case, we shall derivean a posteriori bound on the error between J�u� and J�uh�. The error analyses which we shall perform arebased on two dual problems, the ®rst of which stems from the formal adjoint of the partial differentialoperator while the second exploits the adjoint of the stabilised primal operator: ®nd z in H�X� such that

B�w; z� � J�w� 8w 2 H�X�; �3:21�and ®nd zd in H�X� such that

Bd�w; zd� � J�w� 8w 2 H�X�: �3:22�


Let us assume that each of these problems possesses a unique solution. Clearly, the validity of this as-sumption depends on the choice of the linear functional J��. We have the following general result.

Theorem 12. The dual problems (3.21) and (3.22) give rise to the following error representation formulas,respectively:

J�u� ÿ J�uh� � ÿ��b � m�rÿh ; �zÿ zh��Cÿ � �rh; zÿ zh� ÿ �rh; dLzh�; �3:23a�J�u� ÿ J�uh� � ÿ��b � m�rÿh ; �zd ÿ zd;h��Cÿ � �rh; zd ÿ zd;h� � �rh; dL�zd ÿ zd;h��: �3:23b�

Proof. We shall only present the argument in the case of the stabilisation-dependent dual problem (3.22);for the standard dual problem (3.21) the proof is analogous. On choosing w � uÿ uh in (3.22) and recallingthe Galerkin property (3.8), we deduce that

J�u� ÿ J�uh� � J�uÿ uh� � Bd�uÿ uh; zd� � Bd�uÿ uh; zd ÿ zd;h�� ÿ ��b � m�rÿh ; zd ÿ zd;h�Cÿ � �rh; zd ÿ zd;h� � �drh; L�zd ÿ zd;h��:

That implies (3.23b). �

Guided by the error representation formulas (3.23a) and (3.23b), we de®ne the residual terms q�i�,i � 1; 2; 3, by

q�1� � ÿ�b � m�rÿh ; q�2� � q�3� � rh;

and the weighting terms x�i�, i � 1; 2; 3, as follows:· (a) for the dual problem (3.21):

x�1� � x�2� � zÿ zh; x�3� � ÿdLzh;

· (b) for the stabilisation-dependent dual problem (3.22):

x�1� � x�2� � zd ÿ zd;h; x�3� � dL�zd ÿ zd;h�;for zh and zd;h in Sh. Using this notation, (3.23a) and (3.23b) can be rewritten in the following compact form:

J�u� ÿ J�uh� � �q�1�;x�1��Cÿ � �q�2�;x�2�� q�3�;x�3��:Given a linear functional J�� and a positive tolerance TOL, the aim of the computation is to calculate uh sothat

jJ�u� ÿ J�uh�j6 TOL: �3:24�A su�cient condition for this to hold is that the stopping criterion

E1�uh�6 TOL �3:25�is satis®ed, where

E1�uh� � j�q�1�;x�1��Cÿ � �q�2�;x�2�� q�3�;x�3��j: �3:26�If (3.25) holds true, J�uh� is accepted as an accurate representation of J�u�; otherwise, uh is discarded and anew approximation is computed on a ®ner partition. In order to ensure that the partition is re®ned onlywhere necessary, a local decision has to be made on each element j as to whether diam�j� is acceptable inrelation to TOL. A convenient approach to constructing a re®nement criterion which relates the local el-ement size to TOL is to localise the right-hand side in (3.26). More precisely, E1�uh� is further bounded fromabove by decomposing the inner products over X and Cÿ as sums of inner products over elements j in Xand faces oj in Cÿ, respectively, with the absolute value sign now appearing under the summation sign:

jJ�u� ÿ J�uh�j6E2�uh�; �3:27�


where

E2�uh� �Xj2Th

j�q�1�;x�1��oj\Cÿ � �q�2�;x�2��j � �q�3�;x�3��jj: �3:28�

Thus, for example, a possible re®nement criterion might consist of checking whether on each element j inthe partition Th the following inequality holds:

j�q�1�;x�1��oj\Cÿ � �q�2�;x�2��j � �q�3�;x�3��jj6TOL

N; �3:29�

where N is the number of elements in Th. If (3.29) holds on each element j in Th then, according to (3.27),the required error control (3.24) has been achieved. As a precursor to the numerical experiments in Section4 we remark here that the seemingly harmless transition from (3.26) to (3.28) can be detrimental: degra-dation of the asymptotic properties of the bound and uneconomical meshes may result. In fact, it will beshown in Section 4.4 that the occurrence of such adverse effects (stemming from the localisation of globalerror bounds) is closely related to the choice of the dual problem.

While the local residuals q�i�jj, i � 1; 2; 3, can be easily evaluated once the numerical solution uh has beencomputed, the calculation of the weights x�i�jj, i � 1; 2; 3, requires special care; this will be discussed indetail in the next section. First, however, we perform two `thought experiments' which illuminate thesigni®cance of using the stabilisation-dependent dual problem (3.22) and illustrate the implications of our aposteriori error analysis. We consider the linear advection equation (3.1) on the unit square X � �0; 1�2 inR2, with b � �1; 1�T , c � 0 and f � 0. The in¯ow boundary datum g will be assumed to be such that theexact solution u is smooth, but not piecewise constant or linear.

Thought experiment 1. First, we choose the mean-¯ow across the out¯ow boundary C� as the functionalto be controlled; namely, we take

J�w� � �1; �b � m�w�C� :In this case, the solution z � 1 of the standard dual problem

ÿb � rz � 0 in X; z � 1 on C�;

is also the solution of the corresponding stabilisation-dependent dual problem (3.22). In view of the erroridentity (3.23b), this implies that J�u� � J�uh�, recovering the well known conservation property of thescheme (3.6).

Thought experiment 2. Next, we choose as the functional to be controlled the mean-¯ow over X; thus, wetake

J�w� � �1;w�:The solution of the corresponding standard dual problem

ÿb � rz � 1 in X; z � 0 on C�;

is area-wise linear with an edge along the line y � x. This results in nonzero values for the error indicators�rh; d�b � rzh��j in all of X inducing global mesh re®nement which does not seem necessary for this par-ticular functional, in view of the conservation property of the scheme. Now we slightly change the errorfunctional J�� to

Jd�w� � �1;w� ÿ �1; d�b � m�w�Cÿ ;and assume that the stabilisation parameter d is a constant function over X; then, a simple calculationshows that the corresponding stabilisation-dependent dual problem (3.22) can be rewritten as

�ÿb � rzd;wÿ d�b � rw�� b � m�zd;w�C� � �1;w� ÿ d�1; �b � m�w�Cÿ 8w 2 H�X�;and it has the same area-wise linear solution as the problem

ÿb � rzd � 1 in X; zd � d on C�:


Consequently, the local error indicators j�q�2�;x�2��jj � j�q�3�;x�3��jj are nonzero only along the charac-teristic line y � x. This indicates that mesh re®nement based on the a posteriori error bound which stemsfrom the stabilisation-dependent dual problem will be localised to the line y � x and will be, therefore,highly economical.

De®nition 13. For future reference, we recall the particular forms of the terms on the right-hand sides of(3.23a) and (3.23b), and we de®ne

I1 � j��b � m�rÿh ; zÿ zh�Cÿ j; II1 � j�rh; zÿ zh�j; III1 � j�drh; Lzh�jand

I1;d � j��b � m�rÿh ; zd ÿ zd;h�Cÿ j; II2;d � j�rh; zd ÿ zd;h�j; III1;d � j�drh; L�zd ÿ zd;h��j:We also consider the localised counterparts of these, given by

I2 �Xj2Th

j��b � m�rÿh ; zÿ zh�oj\Cÿ j; II2 �Xj2Th

j�rh; zÿ zh�jj; III2 �Xj2Th

j�drh; Lzh�jj;

and the corresponding terms I2;d, II2;d, III2;d for the stabilisation-dependent dual problem, arising from theright-hand side of (3.28).

By applying the Cauchy±Schwarz inequality on I2, II2 and III2, as well as I2;d, II2;d and III2;d togetherwith the approximation property (B), these terms can be further bounded above. Thus we arrive at thefollowing result.

Lemma 14. Let u and uh denote the solutions of (3.1) and (3.6), respectively, and suppose that the dual so-lution z is sufficiently regular. Then, we have that

I26Ci

Xj2Th

kbkL1�oj\Cÿ�khrÿh kL2�oj\Cÿ� min06 r6 p

fhrjjzjHr�1�oj\Cÿ�g;

II26Ci

Xj2Th

khrhkL2�j� min06r6 p

fhrjjzjHr�1�j�g;

III26Ci

Xj2Th

kbkL1�j��

� hjkckL1�j��kdrhkL2�j� min

06 r6 pfhr

jjzjHr�1�j�g

�Xj2Th

kbkL1�j��

� hjkckL1�j��kdrhkL2�j�jzjH1�j�:

For the case of the stabilisation-dependent dual problem, z is replaced by zd and estimates identical to those forI2 and II2 hold for I2;d and II2;d, while in the corresponding bound for III2;d the second sum on the right-handside of the estimate for III2 does not arise.

We see from Lemma 14 and inequality (3.27) that when the analysis is based on the standard dualproblem, the resulting bound on the error in the approximation of the functional J�� is dominated by theterm III2 which stems from the stabilisation in the numerical method.

We further illustrate the features of our a posteriori error estimation by a simple example which concernsthe ®nite element approximation of the normal ¯ux through the out¯ow boundary C� for the solution tothe transport problem (3.1). Thus, assuming that w 2 L2�C�� is a given weight function, we are interested inthe quantity

Nw�u� �Z

C��b � m�uwds; �3:30�

where u is the solution to (3.1). Supposing that uh is the Galerkin approximation to u de®ned by the sta-bilised ®nite element scheme (3.6), we wish to estimate the error between the computed out¯ow normal ¯ux


Nw�uh� and the actual value Nw�u� by means of the two dual problems. In Section 4 we present a number ofnumerical experiments to compare the two approaches.

For this particular functional the dual problems (3.21), (3.22) in strong form read as follows: ®nd z suchthat

L�z � ÿr � �bz� � cz � 0 in X; z � w on C�; �3:31�for d 2 W 1

1�X�, ®nd zd 2 H�X� such that

L��zd � dLzd� � 0 in X;

Lzd � 0 on Cÿ; �3:32�zd � dLzd � w on C�:

We shall now derive an a priori bound on the error in the computed functional in terms of Sobolev normsof the analytical solution u; this will indicate the expected rate of convergence for jNw�u� ÿ Nw�uh�j as htends to zero. We need the following result which is a minor variation on an a priori error bound from [10].

Lemma 15. Assume that (B) holds and let u and uh denote the solutions of (3.1) and (3.6), respectively.Suppose that u 2 Hs�1�X� for some s, 06 s6 p, and there exist positive constants cd and Cd such thatcdh6 d6Cdh on X; then

jjjuÿ uhjjj6Chs�1=2jujHs�1�X�;

where

jjjvjjj2 � k��dp

Lvk2L2�X� � kvk

2L2�X� � kvk

2L2�oX�

and C is a constant independent of the mesh function h.

Thereby, from Lemmas 14 and 15, we deduce the following a priori error bound.

Lemma 16. Assume that (B) holds and let u and uh denote the solutions of (3.1) and (3.6), respectively. Giventhat u 2 Hs�1�X� for some s, 06 s6 p, the dual solution z is sufficiently regular, and there exist positiveconstants cd and Cd such that cdh6 d6Cdh on X, we have that

I26Ch2s�3=2 jujHs�1�X� jzjHs�1�Cÿ�;

II26Ch2s�1 jujHs�1�X� jzjHs�1�X�;

III26Ch2s�1 jujHs�1�X� jzjHs�1�X� � Ch2 jujHs�1�X� jzjH1�X�;

where C is a constant independent of the mesh function h. For the case of the stabilisation-dependent dualproblem, z is replaced by zd and estimates identical to those for I2 and II2 hold for I2;d and II2;d, while in thecorresponding bound for III2;d the second term on the right-hand side of the estimate for III2 does not arise.

Thus, Lemma 16 indicates that (under suitable assumptions on the smoothness of u and the dual solutionz) the error bound in the outward normal ¯ux based on the standard dual problem is of size O�h2�, irre-spective of the degree p of the polynomial approximation used in the ®nite element space Sh. On the otherhand, using the error estimate based on the stabilisation-dependent dual problem we ®nd thatjNw�u� ÿ Nw�uh�j � O�h2s�1� for 06 s6 p, provided that zd 2 H s�1�X� with kzdkHs�1�X�6Const:, independentof h.

Remark 17. We note that if we assume that u 2 Hs�1�Cÿ� for some s, 06 s6 p, then the a priori errorbound for the boundary term I2 stated in Lemma 16 may be improved provided that the boundary residualis rede®ned as rÿh � g ÿ Phg. Indeed, on selecting zh � Phz in (3.23a) with Ph chosen as indicated at the end ofRemark 7 and applying (B) we deduce that


I26 Ch2s�2 jujHs�1�Cÿ� jzjHs�1�Cÿ�;

provided that z 2 H s�1�Cÿ�. An analogous bound holds for I2;d.

Remark 18. The absence of a forcing function in the ®rst dual problem (3.31) means that the error rep-resentation formula (3.23a) may be re-written in the following form:

Nw�u� ÿ Nw�uh� � ÿ ��b � m�rÿh ; zÿ zh�Cÿ � �rh; zÿ zh� � �drh; L�zÿ zh��ÿ �drh; �c� cÿr � b�z�:

Thus, if the negative-adjoint stabilisation is employed, i.e. L :� ÿL�, then

c� cÿr � b � 0 for all x in �X;

and it is easy to verify that the solutions to the two dual problems (3.31) and (3.32) coincide, i.e. z � zd.Consequently, when L :� ÿL� the error representation formulas resulting from the two dual problems areidentical; in particular, this is the case when the absorption term c is identically zero and the velocity ®eld bis incompressible, i.e. r � b � 0 for all x in X.

4. Computational implementation; numerical experiments

In this section we present a number of numerical experiments to compare the various error bounds whichwere based on the two dual problems, (3.31) and (3.32). In Examples 1±3 of Sections 4.3, Sections 4.4,Sections 4.5, we implemented (3.6) with the streamline di�usion stabilisation, corresponding to c � 0.

4.1. Numerical approximation of the dual solution

First, we outline the numerical method employed for the approximation of the analytical solution zd tothe stabilisation-dependent dual problem (3.32); here, we exploit the stabilised ®nite element method in-troduced and analysed in [7]. Let Thd be an admissible subdivision of X into shape regular ®nite elements j,with corresponding mesh function hd . Further, we de®ne Shd as the ®nite element space consisting ofcontinuous piecewise polynomials of ®xed degree pd , pd P 1; for simplicity, in this paper we shall onlyconsider the case where pd � p � 1.

Let us ®rst rewrite the stabilisation-dependent dual problem (3.32) in the following strong form

Ldzd � ÿr � �adrzd� � bd � rzd � cdzd � 0; �4:1�where

ad � dbbT; bd � �d�cÿ c� ÿ 1�b; cd �L��1� dc�:We emphasize here that the matrix bbT is positive semi-de®nite, but not positive de®nite. At any rate, (4.1)is a second-order partial di�erential equation with nonnegative characteristic form. Now let us introducethe bilinear form Bd;dd ��; �� de®ned by

Bd;dd �w; v� � �adrw;rv� ÿ �w;r � �bdv�� cdw; v� � ��b � m�dcw; v�C�X

j2Thd

�Ldw; dd bd � rv�j ÿ ��b � m�w; v�Cÿ ;

and the linear functional

ldd �v� � ��b � m�w; v�C� ;where, on element j 2Thd , we de®ne Ld by

Ldw � ÿr � �Pj�adrw�� bd � rw� cdw


and Pj signi®es the orthogonal projection in L2�j�� n onto Ppdÿ1�j�� n

. Here, the dual stabilisation pa-rameter dd is chosen as the positive piecewise constant function dd � Cdhd , where hd jj � diam�j� forj 2Thd . The ®nite element approximation to zd is then de®ned as follows (cf. [7]): ®nd zd 2 Shd such that

Bd;dd �zd; vhd � � ldd �vhd � 8vhd 2 Shd : �4:2�Section 4.2 is devoted to the description of the adaptive ®nite element algorithm.

4.2. Adaptive Algorithm

Following from Theorem 12, we have the a posteriori error bounds:

jNw�u� ÿ Nw�uh�j6 �2�uh; z; zh� for dual problem �3:31�;�2;d�uh; zd; zh;d� for dual problem �3:32�;

��4:3�

where as in (3.28),

�2 �Xj2Th

j ÿ ��b � m�rÿh ; �zÿ zh��oj\Cÿ � �rh; zÿ zh�j ÿ �rh; dLzh�jj;

�2;d �Xj2Th

j ÿ ��b � m�rÿh ; �zd ÿ zd;h��oj\Cÿ � �rh; zd ÿ zd;h�j � �rh; dL�zd ÿ zd;h��jj:

With this notation, we now consider the problem of designing a mesh such that

�2;d�uh; zd; zh;d�6 TOL

subject to the constraint that the number N of elements is minimised. Here, we adopt the optimised meshstrategy outlined in [13]. To this end, we de®ne

A�x� � hÿ3X �x�jrh�zd ÿ zd;h� � drhL�zd ÿ zd;h�j;

B�x� � hÿ4Cÿ�x�j�b � m��g ÿ uh��zd ÿ zd;h�j;

where hX and hCÿ are the mesh functions on X \ C� and Cÿ, respectively. After an elementary calculationinvolving a Lagrange multiplier k, we arrive at the optimal mesh size distributions hopt

X �x� and hoptCÿ �x� given

by

hoptX �

2

3kA

� �1=5

; hoptCÿ �

1

4kB

� �1=5

;

where k is the positive root of

2

3k

� �3=5 ZX

A2=5 dx� 1

4k

� �4=5 ZCÿ

B1=5 ds � TOL: �4:4�

For TOL� 1, we anticipate k� 1, so that �1=k�4=5 � �1=k�3=5. Thus, for simplicity, we neglect the

boundary integral term in (4.4); on eliminating k, we then arrive at the following explicit formula for hoptX �x�:

hoptX �x� �

TOLW

� �1=31

A1=5�x� ; where W �Z

XA2=5�x�dx: �4:5�

In order to construct a computational mesh with granularity predicted by (4.5), we employ the red±greenisotropic re®nement strategy. Here, the user must ®rst specify a (coarse) background mesh upon which anyfuture re®nement will be based. A red re®nement corresponds to dividing a certain triangle into four similartriangles by connecting the midpoints of the sides. Green re®nement is only temporary and is used to re-move any hanging nodes caused by a red re®nement. We note that green re®nement is only applied onelements which have one hanging node; for elements with two or more hanging nodes a red re®nement isperformed. Within this mesh modi®cation strategy, elements may also be removed from the mesh (i.e.dere®ned) provided they do not lie in the original background mesh.


For the practical implementation of this adaptive algorithm, the analytical solution zd to the stabilisa-tion-dependent dual problem (3.32) will be approximated as outlined in the previous section. For clarity, wede®ne �2;d � �2;d�uh; zd; zh;d�, where zh;d is the interpolant of zd onto the mesh Th used to calculate the ap-proximation uh to the primal problem (3.1). The mesh Thd for the dual problem (3.32) will be constructedvia the fixed fraction strategy outlined in [13] using the local error indicator khd rhdkL2�j� as in [6], where rhd isde®ned to be the residual of the computed dual solution zd.

4.3. Example 1

In this example we ®rst investigate the order of convergence of the error in the out¯ow normal ¯ux as themesh is uniformly re®ned. To simplify the presentation, we consider a model problem which ensures thatthe solutions to the two dual problems, (3.31) and (3.32), are identical, cf. Remark 18. To this end, we letb � �1� sin�py�; 2�, c � 0 and f � 0 with boundary condition

u�x; y� � 1ÿ y5 for x � 0; 06 y6 1;eÿ50x4

for 06 x6 1; y � 0:

�In addition, we de®ne the weight function w in the functional Nw��, cf. (3.30), by

w � 1ÿ sin2�p�1ÿ y�=2� cos�py=2� for x � 1; 06 y6 1;

1ÿ �1ÿ x�3 ÿ �1ÿ x�4=2 for 06 x6 1; y � 1:

��4:6�

Thus, the true value of the out¯ow normal ¯ux is Nw�u� � 1:8664.In Table 1 we show kuÿ uhkL2�X� and jNw�u� ÿ Nw�uh�j, along with their respective rates of convergence k,

on a sequence of uniform triangular meshes: in each case the mesh is constructed from a uniform N � Nmesh by connecting the bottom±left corner of each mesh square with its top-right corner. Here, we observethat with the stabilisation parameter chosen to be h=4, the L2�X� norm of the error converges like O�h2� andthe error in the functional, Nw�uÿ uh�, converges like O�h3� as expected, cf. Lemma 16 with s � p � 1.However, if we now look at the convergence rates of each of the terms in the error representation formula(3.23a) (or (3.23b)), cf. De®nition 13, then we observe, cf. Table 2, that the ®rst two terms, i.e. terms I1

(� I1;d) and II1 (� II1;d), both converge like O�h4� as h tends to zero. We note that term II1 is super±convergent, since we only expect to observe a convergence rate of O�h3� as h tends to zero, cf. Lemma 16.Thereby, the whole bound is dominated by the third term, which arises as a result of the stabilisationemployed. Furthermore, we note that the same asymptotic rates of convergence are achieved for each of theabove quantities on unstructured quasi-uniform triangular meshes as well; this indicates that the super±convergence of term II1 is robust with respect to mesh distortion.

Thus while classical a priori error analysis, cf. [10], indicates that the stabilisation should be chosen sothat d � O�h� as the mesh is re®ned, in order to obtain optimal error estimates in a mesh dependent norm,cf. Lemma 15, here, our numerical experiments indicate that d should be chosen to be d � O�h2� in order toobtain O�h4� convergence for the error in the outward normal ¯ux. Indeed, in Tables 3 and 4 we repeat theabove numerical experiments with d � 25h2. From Table 3, we observe that the L2�X� norm of the error stillapproaches second order as the mesh is re®ned, while the error in the functional Nw�� is now clearly O�h4�as h tends to zero. Furthermore, from Table 4 we now clearly observe that all three terms in the error

Table 1

Example 1: Convergence of kuÿ uhkL2�X� and jNw�u� ÿ Nw�uh�j with d � h=4

Mesh kuÿ uhkL2�X� k jNw�u� ÿ Nw�uh�j k

17� 17 2:077� 10ÿ3 ÿ 3:419� 10ÿ5 ÿ33� 33 4:343� 10ÿ4 2:26 4:461� 10ÿ6 2:94

65� 65 1:021� 10ÿ4 2:09 5:657� 10ÿ7 2:98

129� 129 2:500� 10ÿ5 2:03 7:107� 10ÿ8 2:99

257� 257 6:208� 10ÿ6 2:01 8:901� 10ÿ9 3:00


representation formula (3.23a) (or (3.23b)) are fourth order convergent as the mesh is re®ned. As before,the same asymptotic behaviour is achieved on unstructured triangular meshes also.

In summary, this numerical experiment indicates that when the quantity of interest is the outwardnormal ¯ux of the solution, the stabilisation parameter d should not be chosen to be O�h� as advocated ongrounds of accuracy by standard a priori error analysis in the mesh-dependent norm jjj � jjj. Indeed, due tothe inherent super-convergence property of term II1 (and II1;d), cf. De®nition 13, the error in the out¯ow¯ux can be reduced to O�h4�, for a piecewise linear approximation uh to u, provided that d � Cdh2.However, a drawback of such a stabilisation is that the resulting system of linear equations becomes in-creasingly dif®cult to solve as h is reduced, unless Cd is chosen suf®ciently large; this, however, is rathercounter±productive, since a large Cd leads to much larger error in the computed functional. Thus, whilejNw�u� ÿ Nw�uh�j is now O�h4�, the constant that multiplies the powers of h is increased; thereby on practicalmeshes, we would still advocate using O�h� stabilisation with a `small' Cd, at the expense of suboptimalconvergence rates in the functional of interest.

4.4. Example 2

In this example we consider a compressible problem in order to highlight the main di�erences betweenusing the standard dual problem (3.31) and the stabilisation-dependent dual problem (3.32). To this end, welet b � �1� x; 1� y�, c � 0 and f � 0 with same boundary conditions as employed in Example 1; i.e.

Table 4

Example 1: Convergence of the terms in the error representation formula with d � 25h2 (cf. De®nition 13)

Mesh I1 k II1 k III1 k

17� 17 2:671� 10ÿ6 ÿ 3:276� 10ÿ6 ÿ 2:837� 10ÿ4 ÿ33� 33 1:358� 10ÿ7 4:30 1:657� 10ÿ7 4:31 1:966� 10ÿ5 3:85

65� 65 7:210� 10ÿ9 4:24 9:945� 10ÿ9 4:06 1:255� 10ÿ6 3:97

129� 129 4:549� 10ÿ10 3:99 6:373� 10ÿ10 3:96 7:873� 10ÿ8 3:99

257� 257 2:883� 10ÿ11 3:98 4:037� 10ÿ11 3:98 4:924� 10ÿ9 4:00

Table 2

Example 1: Convergence of the terms in the error representation formula with d � h=4 (cf. De®nition 13)


17� 17 1:804� 10ÿ6 ÿ 2:511� 10ÿ6 ÿ 3:490� 10ÿ5 ÿ33� 33 1:119� 10ÿ7 4:01 1:568� 10ÿ7 4:00 4:506� 10ÿ6 2:95

65� 65 7:147� 10ÿ9 3:97 1:003� 10ÿ8 3:97 5:686� 10ÿ7 2:99

129� 129 4:552� 10ÿ10 3:97 6:379� 10ÿ10 3:97 7:126� 10ÿ8 2:97

257� 257 2:876� 10ÿ11 3:98 4:026� 10ÿ11 3:99 8:913� 10ÿ9 3:00

Table 3

Example 1: Convergence of kuÿ uhkL2�X� and jNw�u� ÿ Nw�uh�j with d � 25h2


17� 17 8:442� 10ÿ3 ÿ 2:831� 10ÿ4 ÿ33� 33 7:876� 10ÿ4 3:42 1:963� 10ÿ5 3:85

65� 65 1:099� 10ÿ4 2:84 1:252� 10ÿ6 3:97

129� 129 2:504� 10ÿ5 2:13 7:854� 10ÿ8 3:99

257� 257 6:197� 10ÿ6 2:01 4:912� 10ÿ9 4:00


u�x; y� � 1ÿ y5 for x � 0; 06 y6 1;eÿ50x4

for 06 x6 1; y � 0:

�Selecting the weight function w as in Eq. (4.6), we have Nw�u� � 2:4676.

In Table 5 we ®rst investigate the order of convergence of the error in the functional Nw�� as h tends tozero. As in Example 1, we observe that jNw�u� ÿ Nw�uh�j converges like O�h3� with O�h� stabilisation, whilethe L2�X� norm of the error is of second order. In Tables 6 and 7 we show the convergence of each of theterms in the error representation formulas (3.23a) and (3.23b), respectively. Table 6 again shows that thesecond term in the error representation formula (3.23a), i.e. term II1, is super±convergent; here, II1 � O�h4�as h tends to zero. Furthermore, since the two dual problems are no longer equivalent, the third term in theerror representation formula (3.23a) is only expected to converge like O�h2� as h tends to zero, cf. Lemma16. Table 6 indicates that term III1 is also super±convergent. However, as in Example 1, the error in theoutward normal ¯ux is entirely dominated by this term which arises as a result of the stabilisation em-ployed. On the other hand, from Table 7, we see that while the ®rst term, term I1;d, in the error repre-sentation formula based on the stabilisation-dependent dual problem (cf. (3.23b)) is still fourth orderconvergent as the mesh is re®ned, the second term, term II1;d, is now only O�h3� as h tends to zero. Thus, byemploying the stabilisation-dependent dual problem (3.32) the terms in the error representation formulainvolving integrations over the entire computational domain X, i.e. terms II1;d and III1;d, are `balanced'.This will be essential for ensuring that any a posteriori error estimate derived by further bounding the terms

Table 5



17� 17 2:927� 10ÿ3 ÿ 2:957� 10ÿ4 ÿ33� 33 5:195� 10ÿ4 2:49 3:860� 10ÿ5 2:94

65� 65 1:079� 10ÿ4 2:27 4:944� 10ÿ6 2:96

129� 129 2:544� 10ÿ5 2:08 6:257� 10ÿ7 2:98

257� 257 6:260� 10ÿ6 2:02 7:874� 10ÿ8 2:99

Table 6

Example 2: Convergence of the terms in the error representation formula (3.23a) with d � h=4 (cf. De®nition 13)


17� 17 3:308� 10ÿ6 ÿ 3:350� 10ÿ6 ÿ 2:957� 10ÿ4 ÿ33� 33 1:906� 10ÿ7 4:12 2:295� 10ÿ7 3:87 3:864� 10ÿ5 2:94

65� 65 1:166� 10ÿ8 4:03 1:515� 10ÿ8 3:92 4:947� 10ÿ6 2:97

129� 129 7:236� 10ÿ10 4:01 9:742� 10ÿ10 3:96 6:260� 10ÿ7 2:98

257� 257 4:511� 10ÿ11 4:00 6:178� 10ÿ11 3:99 7:872� 10ÿ8 2:99

Table 7

Example 2: Convergence of the terms in the error representation formula (3.23b) with d � h=4 (cf. De®nition 13)

Mesh I1;d k II1;d k III1;d k

17� 17 3:308� 10ÿ6 ÿ 1:501� 10ÿ5 ÿ 2:773� 10ÿ4 ÿ33� 33 1:906� 10ÿ7 4:12 2:041� 10ÿ6 2:88 3:637� 10ÿ5 2:93

65� 65 1:166� 10ÿ8 4:03 2:562� 10ÿ7 2:99 4:676� 10ÿ6 2:96

129� 129 7:236� 10ÿ10 4:01 3:166� 10ÿ8 3:02 5:933� 10ÿ7 2:98

257� 257 4:511� 10ÿ11 4:00 3:919� 10ÿ9 3:01 7:474� 10ÿ8 2:99


on the right-hand side of (3.26) from above, converges at the same asymptotic rate as the true error in thefunctional Nw��. Indeed, in Table 8 we now compare the convergence rates of the a posteriori error bounds�2 and �2;d, cf. (4.3) and (3.27), together with their respective effectivity indices h1 and h2; hereh1 � �2=jNw�u� ÿ Nw�uh�j and h2 � �2;d=jNw�u� ÿ Nw�uh�j. Here, we see that the error bound �2 now onlyconverges at the sub-optimal rate of O�h2� as h tends to zero; this leads to large effectivity indices whichgrow as the mesh is re®ned. Table 9 shows each of the terms in the error bound �2 together with theirrespective rates of convergence, cf. De®nition 13. Here, we see that while term I2 still converges like O�h4� ash tends to zero, and term II2 is third-order convergent, III2 is now only second order convergent, inagreement with Lemma 16.

In contrast, by using the stabilisation-dependent dual problem, the error bound �2;d converges to zero atthe same asymptotic rate as the true error in the functional Nw��, thereby giving rise to small e�ectivityindices of approximate size 1:3±1:6 on all of the meshes considered, cf. Table 8. Moreover, from Table 10we see that all the terms in the error bound �2;d converge to zero at the same rate as those in the errorrepresentation formula (3.23b); i.e. term I2;d is fourth order convergent and both term II2;d and term III2;d

converge like O�h3� as h tends to zero. Finally, we note that as in Example 1, the same asymptotic beha-viour is achieved on unstructured quasi-uniform triangular meshes.

4.5. Example 3

In this ®nal example we consider a strongly compressible linear hyperbolic problem with discon-tinuous in¯ow boundary condition. To this end, we let b � �10y2 ÿ 12x� 1; 1� y�, c � 0 and f � 0.

Table 8

Example 2: Comparison of the a posteriori error bounds �2 and �2;d with d � h=4

Mesh �2 k h1 �2;d k h2

17� 17 3:195� 10ÿ3 ÿ 10:8 3:724� 10ÿ4 ÿ 1:26

33� 33 8:150� 10ÿ4 1:97 21:1 5:058� 10ÿ5 2:88 1:31

65� 65 2:043� 10ÿ4 2:00 41:3 6:678� 10ÿ6 2:92 1:35

129� 129 5:108� 10ÿ5 2:00 81:6 8:622� 10ÿ7 2:95 1:38

257� 257 1:277� 10ÿ5 2:00 162:2 1:279� 10ÿ7 2:75 1:62

Table 9

Example 2: Convergence of the terms in the a posteriori error bound �2 with d � h=4 (cf. De®nition 13)


17� 17 1:006� 10ÿ5 ÿ 7:431� 10ÿ5 ÿ 3:203� 10ÿ3 ÿ33� 33 4:944� 10ÿ7 4:35 9:117� 10ÿ6 3:03 8:159� 10ÿ4 1:97

65� 65 2:850� 10ÿ8 4:12 1:138� 10ÿ6 3:00 2:044� 10ÿ4 2:00

129� 129 1:734� 10ÿ9 4:04 1:422� 10ÿ7 3:00 5:110� 10ÿ5 2:00

257� 257 1:075� 10ÿ10 4:01 1:777� 10ÿ8 3:00 1:278� 10ÿ5 2:00

Table 10

Example 2: Convergence of the terms in the a posteriori error bound �2;d with d � h=4 (cf. De®nition 13)

Mesh I2;d k II2;d k III2;d k

17� 17 1:006� 10ÿ5 ÿ 1:943� 10ÿ4 ÿ 3:427� 10ÿ4 ÿ33� 33 4:944� 10ÿ7 4:35 2:870� 10ÿ5 2:76 4:639� 10ÿ5 2:89

65� 65 2:850� 10ÿ8 4:12 3:912� 10ÿ6 2:88 6:003� 10ÿ6 2:95

129� 129 1:734� 10ÿ9 4:04 5:125� 10ÿ7 2:93 7:634� 10ÿ7 2:98

257� 257 1:075� 10ÿ10 4:01 6:578� 10ÿ8 2:96 1:138� 10ÿ7 2:75


Here, the characteristics enter the computational domain X from three sides of oX, namely from x � 0,y � 0 and x � 1, and exit X through y � 1. Thereby, we may prescribe the following boundary con-dition

Fig. 1. Example 3: (a) Analytical solution to the primal problem; (b) scaled residual term krhkL2�j�=hj on a 65� 65 mesh with d � h=4;

(c) scaled weighting term kzd ÿ zd;hkL2�j�=h2j on a 65� 65 mesh with d � h=4.


u�x; y� �

0 for x � 0; 0:5 < y6 1;1 for x � 0; 0 < y6 0:5;1 for 06 x6 0:5; y � 0;0 for 0:5 < x6 1; y � 0;sin2�py� for x � 1; 06 y6 1;

8>>>><>>>>:the analytical solution to this problem is shown in Fig. 1 (a). We de®ne the weight function w in thefunctional Nw��, cf. (3.30), by

w � sin�px=2� for 06 x6 1; y � 1: �4:7�

Thus, the true value of the outward normal ¯ux is Nw�u� � 0:24650.In Table 11 we investigate the asymptotic behaviour of kuÿ uhkL2�X� and jNw�u� ÿ Nw�uh�j as the mesh

function h tends to zero. Here, we see that on uniform triangular meshes, the convergence rate of the L2�X�norm of the error is k � 0:4 due to the presence of the two discontinuities in the solution, cf. [6]; theconvergence rate of the L2�X� norm of the error on unstructured meshes is identical, so these results havebeen omitted. In contrast, the error in the outward normal ¯ux is O�h3� on the unstructured meshes andO�h4� on uniform meshes. We note that there is a sudden increase in the order of convergence in the error inthe functional on the uniform meshes followed by a decrease. This behaviour is caused by a change of signin the boundary term in the error representation formula (3.23a); indeed, by calculating the convergence

Table 11


Uniform meshes Unstructured meshes

Mesh kuÿ uhkL2�X� k jNw�uÿ uh�j k jNw�uÿ uh�j k

17� 17 1:176� 10ÿ1 ÿ 3:234� 10ÿ4 ÿ 4:814� 10ÿ4 ÿ33� 33 8:951� 10ÿ2 0:39 2:191� 10ÿ5 3:88 1:069� 10ÿ4 2:17

65� 65 6:858� 10ÿ2 0:38 1:264� 10ÿ6 4:12 1:527� 10ÿ5 2:81

129� 129 5:280� 10ÿ2 0:38 3:509� 10ÿ8 5:17 1:847� 10ÿ6 3:05

257� 257 4:010� 10ÿ2 0:40 5:638� 10ÿ9 2:64 1:941� 10ÿ7 3:25

Fig. 2. Example 3: Adaptive algorithm with TOL � 5:0� 10ÿ6 and d � h=4.


rate k between the error in the outward normal ¯ux on the 65� 65 mesh and the 257� 257 mesh gives riseto a convergence rate of k � 3:90.

We remark that the high±order convergence attained for the error in the outward normal ¯ux is at-tributed to the fact that while the residual terms rh and rÿh are clearly large in the vicinity of the discon-tinuities in this problem, cf. Fig. 1(b), the local weighting terms involving the approximation error betweenzd and zd;h are not active within these regions of the computational domain since the dual solution is ex-tremely smooth with `small' support concentrated near the boundary y � 1, cf. Fig. 1(c). This clearlyhighlights the bene®t of keeping the local weighting terms x�i�j , i � 1; 2; 3, inside the a posteriori error bound(4.3), rather than bounding them above by a global constant.

Fig. 3. Example 3: (a) Initial mesh for primal problem with 61 nodes and 96 elements; (b) initial mesh for stabilisation-dependent dual

problem with 137 nodes and 232 elements; (c) primal mesh using optimal mesh strategy (4.5) for TOL � 5:0� 10ÿ5 with 5648 nodes and

11132 elements (jNw�uÿ uh�j � 6:764� 10ÿ6); (d) dual mesh with 7594 nodes and 14199 elements; (e) mesh constructed using energy

error indicator with 8607 nodes and 17038 elements (jNw�uÿ uh�j � 3:057� 10ÿ5).


In Fig. 2 we show the performance of the adaptive algorithm described in Section 4.2 forTOL � 5:0� 10ÿ6; here, the initial meshes used for the numerical solution of the primal problem and thestabilisation-dependent dual problem are shown in Fig. 3(a) and (b), respectively. We clearly observe inFig. 2 that even though the solution to the stabilisation-dependent dual problem has been numericallyapproximated, the a posteriori error estimator �2;d remains an upper bound on the true error in the func-tional Nw��; for clarity, we write Nw�e��2;d�� to denote the error in the outward normal ¯ux on the sequenceof meshes generated using the optimised mesh strategy (4.5). Additionally, in Fig. 2 we plot the true error inthe functional Nw�� based on a sequence of meshes generated using the energy error indicator krhkL2�j� oneach element j in the mesh Th with the ®xed fraction re®nement strategy, cf. [13]; we denote this byNw�e�rh��. Here, we see that Nw�e�rh�� is always inferior to Nw�e��2;d��; moreover, on the sequence of meshesgenerated by using the energy error indicator, the error in the outward normal ¯ux starts to `level o�' as thenumber of degrees of freedom in the mesh increases. This indicates that by using an ad hoc mesh re®nementstrategy only a limited amount of accuracy may be achieved in the functional of interest.

In Fig 3 (c) and (d) we show the meshes generated for the primal and dual solution, respectively, usingthe adaptive algorithm outlined in Section 4.2 for TOL � 5:0� 10ÿ5. Here, the mesh for the primal problemis concentrated in the upper region of the domain close to the out¯ow boundary. Most notable is the lack ofmesh re®nement in the discontinuities as they enter the domain from y � 0 and x � 0. The mesh constructedfor the stabilisation-dependent dual problem is again denser in the upper half of X, with most of the ele-ments concentrated near the corners �x; y� � �0; 1� and �x; y� � �1; 1� due to the presence of a boundarylayer of thickness � jb � mjd in the dual solution zd along Cÿ; here b denotes the normalised velocity vector.Finally, in Fig. 3(e) we show the mesh constructed using the L2�j� norm of the residual; as expected most ofthe elements are concentrated in the discontinuities, leading to unnecessary over-re®nement.

5. Concluding remarks

In this article we have developed the a posteriori error analysis of stabilised ®nite element approxima-tions to transport problems via duality arguments. In particular, by using a stabilisation-dependent dualproblem which respects the particular structure of the Galerkin method employed, optimal error boundsfor both norms and linear functionals of the approximation error were established. In the context of es-timating the outward normal ¯ux of the solution, we have implemented an adaptive algorithm capable ofdelivering guaranteed error control to within a user±de®ned tolerance TOL. Here, mesh adaptivity wasemployed for the numerical estimation of both the primal and dual problems; for the former problem themesh was constructed using the optimised mesh strategy outlined in [13].

The present theory extends directly to multi-dimensional symmetric positive systems in the sense ofFriedrichs (cf. [6,15]). Further extensions to nonlinear hyperbolic conservation laws, based on a suitablelinearisation of the primal problem are possible; this work is part of our current research programme andwill be presented elsewhere.

Acknowledgements

Paul Houston and Endre S�uli acknowledge the ®nancial support of the EPSRC (Grant GR/K76221).Rolf Rannacher acknowledges the ®nancial support of the German Research Association (DFG) throughthe SBF 359 ``Reactive Flow, Di�usion and Transport'' at the University of Heidelberg.

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A posteriori error analysis for stabilised finite element approximations of transport problems