Contents Introduction 3 Johnson's paradigm 4 3 A posteriori error analysis for steady hyperbolic PDEs 7 3. Symmetric hyperbolic systems : : : : : : :

Transcription

1 Report no. 96/09 Finite element methods for hyperbolic problems: a posteriori error analysis and adaptivity a Endre Suli Paul Houston This paper presents an overview of recent developments in the area of a posteriori error estimation for rst-order hyperbolic partial dierential equations. Global a posteriori error bounds are derived in the H norm for steady and unsteady nite element and nite volume approximations of hyperbolic systems and scalar hyperbolic equations. We also consider the problem of a posteriori error estimation for linear functionals of the solution. The a posteriori error bounds are implemented into an adaptive nite element algorithm. Subject classications: AMS(MOS): 65M5, 65M50, 65M60 Key words and phrases: a posteriori error analysis, adaptivity, hyperbolic problems The rst author would like to acknowledge the nancial support of the EPSRC. The work reported here forms part of the research programme of the Oxford{Reading Institute for Computational Fluid Dynamics. Oxford University Computing Laboratory Numerical Analysis Group Wolfson Building Parks Road Oxford, England O 3QD May, 996 a Paper presented as Invited Lecture at the State of the Art in Numerical Analysis Conference, in York, -4 April 996.

2 Contents Introduction 3 Johnson's paradigm 4 3 A posteriori error analysis for steady hyperbolic PDEs 7 3. Symmetric hyperbolic systems : : : : : : : : : : : : : : : : : : : : 7 3. A posteriori bound on the global error : : : : : : : : : : : : : : : 3.3 Application to the cell vertex nite volume method : : : : : : : : 4 A posteriori error estimation for linear functionals 4 5 Unsteady problems 9 5. A posteriori error analysis : : : : : : : : : : : : : : : : : : : : : : 5.. Error representation : : : : : : : : : : : : : : : : : : : : : 5.. Interpolation/projection estimates for the dual problem : : 5..3 Strong stability of the dual problem : : : : : : : : : : : : : Completion of the proof of the a posteriori error estimate : 4 5. Computational implementation : : : : : : : : : : : : : : : : : : : Adaptive algorithm : : : : : : : : : : : : : : : : : : : : : : Mesh modication strategy : : : : : : : : : : : : : : : : : Numerical experiments : : : : : : : : : : : : : : : : : : : : : : : : 8 6 Conclusions 9 References 3

3 3 Introduction Partial dierential equations of hyperbolic type are of fundamental importance in many areas of applied mathematics, particularly uid dynamics and electromagnetics. Solutions to these equations exhibit localised phenomena, such as propagating discontinuities and sharp transition layers, and their reliable numerical approximation presents a challenging computational task; indeed, in order to resolve such localised features in an accurate and ecient manner it is essential to use adaptively rened computational meshes whose construction is governed by sharp a posteriori error bounds. Over the last decade the a posteriori error analysis of nite element methods for partial dierential equations of hyperbolic and nearly-hyperbolic character has been a subject of intensive research. For an overview of current activity in this area we refer to the article of Johnson []; see also, [0], [], [3], and references therein. The approach proposed in those papers, and reviewed in the next section, rests on performing an elliptic regularisation of the hyperbolic problem and exploiting the smoothing properties of the resulting adjoint problem in conjunction with Galerkin orthogonality. Subsequent sections of the paper are devoted to discussing an alternative, more direct, approach to the a posteriori error analysis of nite element approximations of hyperbolic problems which avoids the need for regularisation. Because of the inherent lack of strong smoothing properties in hyperbolic problems, particular care has then to be taken about existence of traces and integrationby-parts formulae which the a posteriori error analysis relies on; the necessary functional-analytic prerequisites that concern these issues are reviewed in the third section. By exploiting this theory, we derive a posteriori bounds on the global error in the H norm for a general class of nite element methods (including certain nite volume schemes, interpreted as Petrov-Galerkin methods) for symmetric hyperbolic systems. For related work, see [5], [6], [7], [], [5]. Controlling the computational error in a given norm, however, is not always the ultimate goal in practice. Indeed, in many engineering problems the accurate approximation of particular linear functionals of the solution (such as the lift, the drag, and the ux across the boundary of the domain) is at least as important as the reliable approximation of the solution in a given norm over the entire computational domain. We show, by considering the example of estimating the normal ux through the boundary, how problems of this kind may be approximated in an ecient way by exploiting a hyperbolic duality argument in conjunction with theoretical results concerning the propagation of singularities. In the nal part of the paper we discuss unsteady hyperbolic problems. In particular, we consider the a posteriori error analysis of a class of evolution- Galerkin methods for multi-dimensional scalar hyperbolic equations on a spatial domain and nite time interval [0; T ]. Here, we derive an a posteriori bound on the global error in the norm of the space L (0; T ; H ()). We conclude by indicating the practical relevance of this work and by illustrating the implementation

4 4 of the a posteriori analysis into an adaptive evolution-galerkin algorithm. Johnson's paradigm In this brief section we review, in the context of rst-order hyperbolic systems, the general theoretical framework of a posteriori error estimation pursued by Johnson and his co-workers; for a comprehensive account, see [] and [3]. Suppose that M is a Hilbert space with inner product (; ) and let A : M! M be a linear operator on M with domain D(A) M and range R(A) = M (in our case a system of rst-order hyperbolic dierential operators). Given that f M, consider the problem of nding u D(A) such that Au = f: In order to construct a Galerkin approximation to this problem, we consider a sequence of nite-dimensional spaces fu h g, parameterised by the positive discretisation parameter h; for the sake of simplicity we shall suppose that U h D(A) for each h. Simultaneously, consider a sequence of nite-dimensional spaces fm h g, with M h contained in M for each h. For the purposes of this paper, U h and M h can be thought of as standard nite element spaces consisting of piecewise polynomial functions on a partition, of granularity h, of the computational domain. Let h denote the orthogonal projector in M onto M h. The Galerkin approximation u h of u is then sought in U h as the solution of the nite-dimensional problem h Au h = h f: In order to obtain a computable bound on the global error e h = u u h in terms of the residual r h, dened by r h = f Au h ; we begin by observing the Galerkin orthogonality property (r h ; v h ) = 0 8v h M h : This will be a key ingredient in the analysis. In addition, denoting by A the adjoint of A, we consider the following auxiliary problem, referred to as the dual problem: A = u u h : The a posteriori error analysis is based on a duality argument, and it proceeds as follows: ku u h k = (u u h ; u u h ) = (u u h ; A ) = (A(u u h ); ) = (Au Au h ; ) = (f Au h ; ) = (r h ; ):

5 5 By virtue of the Galerkin orthogonality property (r h ; h ) = 0 for any h M h, and therefore ku u h k = (r h ; h ) = (h s r h ; h s ( h )); where s is a non-negative real number, to be chosen below. By virtue of the Cauchy-Schwarz inequality, ku u h k kh s r h k kh s ( h ))k: While the rst term on the right-hand side is of the desired form, involving the (computable) residual r h multiplied by an appropriate power of the discretisation parameter, the second term incorporates, the solution to the dual problem. Since the dual-problem has the global error as data, is unknown and has to be eliminated from the analysis by relating it to u u h ; furthermore a suitable choice of h has to be made. This is achieved as follows: suppose that fw g 0 is a scale of Hilbert spaces, with associated norms jjj jjj, such that W 0 = M and W is continuously embedded into W whenever. Furthermore, we assume that there exist h M h and a positive constant C z such that, for W s, kh s ( h )k C z jjjjjj s : For nite element methods, this hypothesis is easily fullled by choosing W s as a hilbertian Sobolev space of index s and referring to standard approximation properties of piecewise polynomial functions in Sobolev spaces, with h taken as the projection, the interpolant or the quasi-interpolant of from M h. Thereby we arrive at the bound ku u h k C z kh s r h k jjjjjj s : This brings us to the nal, and most important, step in the a posteriori error analysis. The norm jjjjjj s appearing on the right-hand side of the last inequality has to be eliminated in terms of u u h by recalling the relationship between and u u h, namely that A = u u h. Thus to proceed, we assume that A is invertible and that (A ) is a bounded linear operator from M to W s ; hence jjjjjj s = jjj(a ) (u u h )jjj s C? ku u h k; where C? is a positive constant, greater than or equal to the norm of (A ). Finally, combining the last two bounds we arrive at the desired a posteriori bound on the global error e h = u u h in terms of the computable residual r h : ku u h k C z C? kh s r h k: In order to turn this into an error bound that can be used in practice, the constants C z and C? have to be determined. Estimating C z is a relatively simple matter using readily available results from approximation theory (see, for

6 6 example, Exercise 3.. in Ciarlet's monograph [5] for an explicit formula for C z in the case of standard nite element spaces consisting of continuous piecewise polynomials on simplices, or the work of Handscomb [7] for very much sharper estimates of C z for piecewise linear nite elements on triangles). On the other hand, supplying C? is much harder, involving an analytical study of the wellposedness of the dual problem. Since any value of C? that is arrived at through such general analytical arguments is necessarily a considerable overestimate of the ratio jjjjjj s =ku u h k, in practice the stability constant C? is determined computationally for the problem at hand, as part of the process of a posteriori error estimation. Finally, we have to x s, the exponent of h in the error bound. As one would like the a posteriori bound to reect the approximation property of the space M h to its full extent, one would wish to choose s as large as possible; unfortunately, for linear rst-order hyperbolic systems consisting of m equations on a domain R d, (A ) is a bounded operator from M = [L ()] m to M = W 0 = [L ()] m only, so s cannot exceed 0; thus we end up with the bound ku u h k C z C? kr h k: In fact, we note that when s = 0 there is no benet in exploiting Galerkin orthogonality, and we may simply take h = 0 in our argument to improve this bound to ku u h k C? kr h k: One way or the other, in the case of a rst-order hyperbolic system, the a posteriori error bound that we arrive at on the basis of the reasoning outlined above is unsatisfactory in that it fails to display the approximation properties of the test space M h. Worse still, when the data are discontinuous, linear hyperbolic systems may possess solutions that are discontinuous across characteristic hypersurfaces and, under mesh renement, kr h k will then converge to 0 very slowly, if at all; consequently, in the absence of the compensating factor h s, any adaptive algorithm driven by this error bound is likely to be inecient. The problem may be rectied by perturbing the rst-order hyperbolic operator A (or, indeed, both A and A ) through the addition of a second-order elliptic term with small coecient; this then provides additional regularity which, in favourable circumstances, allow one to take s =. This approach, however, is associated with undesirable complications related to the fact that articial boundary conditions have to be supplied for the resulting second-order operator in such a way that the features of the solution to the hyperbolic system are retained in the vicinity of the boundary; a further diculty with elliptic regularisation of non-dissipative hyperbolic systems, such as the Maxwell system of electro-magnetism, is that it may introduce a physically unacceptable level of damping into the model. Thus, in the rest of the paper, we consider an alternative approach.

7 3 A posteriori error analysis for steady hyperbolic PDEs In this section we develop the a posteriori error analysis of a general class of nite element methods for steady linear multi-dimensional hyperbolic systems. We begin with a review of basic results from the theory of symmetric hyperbolic systems in the sense of Friedrichs. 3. Symmetric hyperbolic systems Throughout this section will denote a bounded open set in R d with Lipschitz continuous boundary. Suppose that A i, i = ; :::; d, and C are (matrixvalued) mappings from into R mm, m ; we shall assume that, for each i, the entries of A i are continuously dierentiable on and the components of C are continuous on. We consider the hyperbolic system of linear rst-order partial dierential equations Lu d i= x i (A i u) + Cu = f in. (3.) The system (3.) is said to be symmetric positive if the following conditions hold: (a) the matrices A i, for i = ; :::; d, are symmetric, i.e. A i = A i ; (b) there exists 0 and a unit vector R d, such that the symmetric part of the matrix K = C + d i= A i x i + d i= i A i is positive denite, uniformly on, i.e. there exists a positive constant c 0 = c 0 () such that for all x in. (K (x) + K (x)) c 0 I (3.) Since is Lipschitz continuous, the outer unit normal vector eld ^ = (^ ; :::; ^ d ) is dened almost everywhere on with respect to the (d )- dimensional measure on. Consider the symmetric matrix B = ^ A + ::: + ^ d A d : We shall suppose that B is non-singular almost everywhere on, that is, the boundary of is non-characteristic for L. Since B is symmetric and of full rank it can be decomposed as B = B + + B, where B + is positive semi-denite and B 7

8 8 is negative semi-denite. Given that g is a suciently smooth function dened on, an admissible boundary condition for (3.) is given by B u = B g on ; (3.3) we shall also consider the homogeneous counterpart of this boundary condition: B u = 0 on : (3.4) At this stage these boundary conditions are to be understood formally; below we shall state a trace theorem which assigns a precise meaning to B uj. First, however, we dene the graph space of the operator L as the linear space When equipped with the norm H(L; ) = fv [L ()] m : Lu [L ()] m g: jjjvjjj ; = ke (x) vk + ke (x) Lvk the graph space H(L; ) is a Hilbert space. Here and throughout the paper k k will stand for the L norm over induced by the L () inner product (; ). Consider L, the formal adjoint of L, dened by L v = d i= A i v x i + C v; the associated graph space H(L ; ) and graph norm jjj jjj ;; are dened analogously as for L, but with the weight-function e (x) replaced by e (x). Let 0; : [H ()] m! [H = ()] m signify the usual trace operator (which to each element of [H ()] m assigns its restriction to ). Let [H = ()] m denote the dual space of [H = ()] m. The following trace theorem will play a key r^ole in our error analysis (see [7]). Theorem 3. Assuming that is non-characteristic for the operator L, the mapping B; : v 7! B( 0; v) dened on [H ()] m can be extended by continuity to a linear and continuous mapping, still denoted B;, from H(L; ) into [H = ()] m. Moreover, for any u H(L; ) and v [H ()] m the following Green's formula holds (Lu; v) (u; L v) = h B; u; 0; vi: An analogous result is valid for H(L ; ). Here and throughout the rest of this paper, h; i denotes the duality pairing between the function spaces [H = ()] m and [H = ()] m.

9 9 The splitting B = B + + B induces a natural decomposition of the trace operator B; which leads to the denition of the partial trace operators B ; : H(L; )! [H = ()] m with Indeed, letting B; = B+ ; + B ; : B ; u = B ( 0; u) 8u [H ()] m ; on the dense subspace [H ()] m of H(L; ) this decomposition can be constructed by splitting the trace of u [H ()] m ; the linear operators B ; : H(L; )! [H = ()] m are then dened by continuous extension from the dense subspace [H ()] m to all of H(L; ). One can proceed in the same way for H(L ; ). Thus, in the case of the homogeneous boundary value problem (3.), (3.4), we can dene precisely the domains of the operators L and L : D(L; ) = fu H(L; ) : B ; u = 0 on g; D(L ; ) = fu H(L ; ) : B+ ;u = 0 on g; when equipped with the associated graph-norms jjjjjj ;, jjjjjj ;;, D(L; ) and D(L ; ) are Hilbert subspaces of H(L; ) and H(L ; ), respectively. Given that f [L ()] m, a function u [L ()] m satisfying (u; L ) = (f; ) 8 D(L ; ) \ [H ()] m is called a weak solution of the homogeneous boundary value problem (3.), (3.4). A weak solution belonging to H(L; ) is called a strong solution. We note that the requirement that u be a strong solution does not preclude the possibility of u being discontinuous; indeed, since only Lu [L ()] m, discontinuities in u may occur across characteristic hypersurfaces. Theorem 3. Assuming that is a non-characteristic hypersurface for L, and given that f [L ()] m, the homogeneous boundary value problem (3.), (3.4) has a unique strong solution u D(L; ). In addition the linear operator L is a continuous bijection from D(L; ) onto [L ()] m with a continuous inverse L : [L ()] m! D(L; ). An analogous result holds for L. Proof The proof is based on Banach's Closed Range Theorem. Given that v [H ()] m, upon taking the inner product of Lv with e (x) v, integrating by parts using Theorem 3. and splitting the trace operator B;, we have that he (x) B+ ; v; e (x) 0; vi + ( (K + K )e (x) v; e (x) v) = he (x) B ; v; e (x) 0; vi + (e (x) Lv; e (x) v):

10 0 Thence, recalling (3.) of hypothesis (b), we deduce the following Garding inequality: c 0 k e (x) vk k e (x) Lvk 8v [H ()] m \ D(L; ): (3.5) Noting that [H ()] m is dense in H(L; ), it follows that [H ()] m \ D(L; ) is dense in D(L; ), so that ke (x) Lvk c jjjvjjj ; 8v D(L; ); (3.6) where c = ( + c 0 ) =. Now (3.6) implies that L is an injective operator from D(L; ) onto its range space R(L), and that the inverse of L is continuous. Hence L is an isomorphism from D(L; ) onto R(L); this implies that R(L) is a closed linear subspace of [L ()] m. By virtue of hypothesis (b), the transpose t L of L has trivial kernel, i.e. Ker( t L) = f0g. The Closed Range Theorem then implies that R(L) = Ker( t L)? = [L ()] m. Hence L is an isomorphism from H(L; ) onto [L ()] m. We deduce from this theorem that (weak and strong) solutions of the homogeneous boundary value problem (3.), (3.4) satisfy the stability estimate kuk c 0 e L kfk; where L = diam(). More generally, consider the non-homogeneous boundary value problem (3.), (3.3), where f [L ()] m and g [L ()] m. A function u [L ()] m obeying (u; L ) + hb g; i = (f; ) 8 D(L ; ) \ [H ()] m is called a weak solution of the boundary value problem. A weak solution u of the non-homogeneous problem (3.), (3.3) which belongs to H(L; ) is called a strong solution. Lax and Phillips [4] proved, under the assumption that is noncharacteristic for L, that every weak solution is a strong solution. Furthermore, assuming that f [H ()] m, g [H ()] m and the entries of C are continuously dierentiable on, the following regularity result holds: kuk H () c kfkh () + kgk H () (see Lax and Phillips [4]). An analogous bound holds for the problem L = in ; B + = B + on ; with corresponding stability constant c 0 ; namely, kk H () c 0 (kk H () + kk H ()): (3.7)

11 3. A posteriori bound on the global error Suppose that U h H(L; ) is a nite element trial space consisting of piecewise polynomial functions on a partition ( i ) i=;:::;nh of ; here h is the mesh parameter. Further, let M h [L ()] m be a nite element test space on this partition, and let h denote the orthogonal projector in [L ()] m onto M h. We consider the following general class of Galerkin approximations of the boundary value problem (3.), (3.3): nd u U h such that h Lu h = h f on ; B u h = B g; (3.8) where, for the sake of simplicity, we have assumed that the function g belongs to the restriction of the nite element trial space U h to the boundary. We shall suppose that (3.8) has a unique solution. Particular examples of such discretisations, including nite element and nite volume methods, will be considered below. We denote by e h = u u h the global error and dene the residual r h = f Lu h. Since L is a linear operator, it follows that e h is the unique solution of the boundary value problem Le h = r h on, B e h = 0: This is a key relationship between the (computable) residual r h and the global error e h which allows us to obtain computable bounds on e h in terms of r h by bounding the inverse of the operator L. We shall further suppose that the nite element test space M h possesses the following standard approximation property: (c) There exists a positive constant c, independent of h, such that for each [H ()] m there is h M h with kh ( h )k L ()! c jj H (): Theorem 3.3 Suppose that hypotheses (a), (b) and (c) hold, and that the entries of C are continuously dierentiable on. Then, ku u h k H () c 0 c khr h k L ()! : Proof Given [C 0 ()] m, consider the dual problem L = on ; B + = 0: Then, (u u h ; ) = (u u h ; L ) = (L(u u h ); ) = (r h ; ):

12 Recalling (3.8), it follows that so, for each h M h, we have that (r h ; h ) = 0; (u u h ; ) = (r h ; h ) c khr h k L () khr h k L ()!! According to the regularity result (3.7), with =, jj H () kk H () c 0 k k H (): kh ( h )k L ()! jjh (): (3.9) Substituting this into (3.9), dividing both sides by k k H (), and taking the supremum over all [C 0 ()] m, we obtain the desired error bound. 3.3 Application to the cell vertex nite volume method In this section we present an a posteriori error bound for a cell vertex nite volume approximation of (3.), (3.3); this is a generalisation of a similar result derived, in the case of homogeneous boundary conditions, in [5]. For the sake of simplicity we suppose that = (0; ), and consider the Friedrichs system in conservation form: Lu i= x i (A i u) + Cu = f in ; B uj = B g on : The cell vertex nite volume approximation of this boundary value problem is obtained by subdividing using a structured mesh that consists of convex quadrilateral elements, and integrating the dierential equation over each element; using Gauss' theorem, integrals over cells are converted into contour integrals which are then approximated by the trapezium rule. This construction gives rise to a conservative four-point nite dierence scheme, called the cell vertex scheme. Equivalently, the cell vertex scheme can be formulated as a Petrov-Galerkin nite element method based on continuous piecewise bilinear trial functions and piecewise constant test functions (see [], [3], [4], []). It is this latter interpretation that is adopted here. Let F = ft h g, h > 0, be a family of structured partitions T h of = (0; ) into convex quadrilateral elements. We shall assume that the partition is quasiparallel in the sense that, for each element ij T h, the distance between the midpoints of the diagonals is bounded by c? meas( ij ), where c? is a xed positive

13 3 constant. In order to introduce the relevant nite element spaces, we dene the reference square ^ = ( ; ), and denote by F ij the bilinear function that maps ^ onto the `nite volume' ij. Let Q (^) be the set of bilinear functions on ^, and Q 0 (^) the set of constant functions on ^. We dene M h = n q [L ()] m : q = ^q F ij ; ^q [Q 0 (^)] m ; ij T h o ; U h = n v [H ()] m : v = ^v F ij ; ^v [Q (^)] m ; ij T h o ; as well as U h, the set of all functions v in U h such that B vj = B g on ; here we have assumed, for the sake of simplicity, that g belongs to the restriction of the nite element trial space U h to the boundary of the domain. Let us denote by h : [L ()] m! M h the L -orthogonal projector onto M h, and by I h : (H(L; ) \ [C 0 ()] m )! U h U h the interpolation projector onto U h U h. The cell vertex nite volume approximation is dened as follows: nd u h U h satisfying (div I h (Au h ); q h ) + (Cu h ; q h ) = (f; q h ) 8q h M h ; (3.0) where A = (A ; A ). We dene the cell vertex operator L h : U h! M h by L h : v h 7! h (div I h (Av h )) + h (Cv h ): With this denition, we can reformulate (3.0) as follows: nd u h U h satisfying L h u h = h f: We shall adopt the following hypothesis: (b') The matrices A i, i = ;, are positive-denite, uniformly on. Clearly (b') is sucient for (b) when d =. Under this assumption U h and M h have the same dimension and L h is then easily seen to be bijective. For theoretical results concerning the stability and convergence of the cell vertex scheme we refer to [],[8], [9], [], [3] and [4]. Here we derive an a posteriori error bound for the method. Theorem 3.4 Suppose that hypothesis (b') holds, that F = ft h g is a family of structured quasi-parallel partitions of, and that, given s, < s 3, the coecients of the matrices A i, i = ;, belong to H s () \ C () and those of C to C (). Then, the cell vertex scheme obeys the following a posteriori error bound: ku u h k H () c 3 where c 3 is a computable constant. khr h k L () + jh s Au h j H s ()! ;

14 4 Then, Proof Suppose that [C 0 ()] m and consider the dual problem L = on ; B + = 0: (u u h ; ) = (u u h ; L ) = (L(u u h ); ) = (f Lu h ; h ) + (f Lu h ; h ) = (r h ; h ) + ( h (div I h (Au h ) div (Au h )); h ): Thus, choosing h = h and exploiting the approximation property (c), we have that j(u u h ; )j! k h (div I h (Au h ) div (Au h ))k L () kk! jjh (): (3.) +c khr h k L () By virtue of a superconvergence result stated in [] (for the special case of a uniform nite dierence mesh; the extension to quasi-parallel meshes is straightforward), k h (div I h (Au h ) div (Au h ))k L () c 4 jh s Au h j H s (); < s 3: Inserting this into (3.), recalling the hyperbolic regularity estimate (3.7), jj H () kk H () c 0 k k H (); dividing both sides of the inequality by k k H () and taking the supremum over all in [C 0 ()] m, we obtain the desired result, with c 3 = c 0 max (c ; c 4 ). 4 A posteriori error estimation for linear functionals In engineering applications it is frequently the case that linear functionals of the solution (such as the lift, the drag, or the ux of the solution across the boundary) are of prime concern. In such instances it is unlikely that a posteriori error bounds of the kind stated in the last two theorems will be of use in the design of adaptive algorithms. Our aim in this section is to propose an approach to deriving a posteriori error bounds on linear functionals, without attempting to obtain an upper bound on the error in a norm in which the functional is bounded. In order to illustrate the key ideas we discuss a specic example: the a posteriori estimation of the error in the outow ux across the boundary (or part of the boundary) for a symmetric hyperbolic system. Let us consider the non-homogeneous boundary value problem (3.), (3.3), where f [L ()] m and g [L ()] m. For the sake of simplicity, we assume

15 5 that g belongs to the restriction of the trial space U h to the boundary, and we approximate the non-homogeneous problem by the following method: nd u h in U h such that h Lu h = h f in ; B u h = B g on : Given any [L ()] m, consider the linear functional N (v) = Z B + v ds; v [H ()] m ; here plays the r^ole of a weight function that can be chosen freely (e.g. can be taken to have its support contained in a subset of, etc.). We seek to approximate the (weighted) outow ux N (u) by N (u h ). Theorem 4. Suppose that hypotheses (a), (b) and (c) hold, and that the entries of C are continuously dierentiable on. Then, for each [H ()] m, assuming that u and u h belong to [H ()] m, we have that jn (u) N (u h )j c 0 c = khr h k L ()! k k H (): Proof Consider the dual problem L = 0 in ; B + = B + on : Since u u h is in D(L; ) \ [H ()] m, Green's formula from Theorem 3. gives 0 = (u u h ; L ) = (L(u u h ); ) N (u u h ): Consequently, for any h M h, N (u) N (u h ) = (r h ; h ): Exploiting hypothesis (c), it follows that jn (u) N (u h )j c = khr h k L ()! jj H (); and therefore, by the hyperbolic regularity theorem jn (u) N (u h )j c 0 c = khr h k L ()! k k H ():

16 6 A particularly relevant case concerns the estimation of the ux through an open subset. In this case it is tempting to take = where is the characteristic function of ; unfortunately such does not belong to [H ()] m so Theorem 4. does not apply. We shall illustrate on a simple example of a scalar hyperbolic problem how a rened analysis may be pursued to obtain a bound on the ux across by recalling that singularities propagate along (bi-)characteristics. Consider the scalar hyperbolic equation Lu r (au) + cu = f in (4.) for the unit square (0; ), where a = (a ; a ) is a vector function with positive, continuously dierentiable entries dened on, c is a continuous function of, and f L (). Let b = ^ a + ^ a, where ^ = (^ ; ^ ) is the unit outward normal vector eld to the boundary of ; at the corners of the square we dene ^ to be the outward pointing unit vector collinear with the diagonal passing through that corner. Clearly b is non-zero almost everywhere on, i.e. is non-characteristic for L. We consider the partial dierential equation (4.) subject to the homogeneous boundary condition b u = 0 on ; (4.) here b = min(b; 0) denotes the negative part of b. The boundary condition (4.) is equivalent to requiring that u = 0 along the inow part of the boundary, = fx = (x ; x ) : a(x) ^(x) < 0g; The complement of the inow boundary, + = fx = (x ; x ) : a(x) ^(x) 0g; is called the outow part of the boundary. Let us suppose that is a connected, relatively open measurable subset of the outow boundary +, and that we wish to approximate the normal outow ux along, dened as N (u) = Z + (a ^)u ds; u H () \ D(L; ); where =, and is a xed element in H () which plays the r^ole of a weight function (e.g. on ). Now Sobolev's embedding theorem implies that is continuous on, and therefore has possible discontinuities only at the points and, say, that comprise the boundary of. Consider the characteristic curves C and C of L in that contain the points and, respectively; these two curves divide into three disjoint regions, one of which, denoted 0, has as

17 7 its outow boundary, and two other regions over each of which, the solution of the dual problem, L a r + c = 0 in j + = on + ; is identically zero. Let ;h be a characteristic strip of cross-section c 7h= that contains the characteristic C in its interior, with ;h dened analogously, so that the elements in the partition of that are in contact with C (respectively C ) are contained in ;h (respectively ;h ). We dene ;h (respectively ;h ) as the intersection of the closure of ;h (respectively ;h ) with +. Theorem 4. Let u and u h belong to H () \ D(L; ), and assume that the nite element test space possesses the following approximation property: (c') for each L () that is piecewise H on, there exists h M h and a positive constant c 5, independent of h, such that kh k ( h )k L () c 5 jj H k (); where k = 0 if L () and k = if H (). Then we have the following error bound: 0 jn (u) N (u h )j c 5 c 6 c 7 +c 5 c 6 0 ;h [ ;h kh = r h k L () 6 ;h [ ;h khr h k L () A A k k L() k k H (); where c 5, c 6 and c 7 are positive constants, independent of h, and signies any nite element in the partition of whose closure has nonempty intersection with 0, the domain of inuence for. Proof Since 0 in the complement of 0, it will only be necessary to consider those elements in the partition of the domain whose closure has nonempty intersection with 0. In order to simplify the notation we shall implicitly assume throughout the proof that, for each element that we refer to, \ 0 is non-empty. Arguing in the same manner as in Theorem 4., we deduce that jn (u) N (u h )j = j(r h ; h )j j(r h ; h ) j = j(r h ; h ) j + j(r h ; h ) j ;h [ ;h 6 ;h [ ;h kr h k L ()k h k L () ;h [ ;h

18 8 + khr h k L ()kh ( h )k L () 6 ;h [ ;h 0 c 5 A kk L ( ;h [ ;h ) +c 5 0 ;h [ ;h kr h k L () 6 ;h [ ;h khr h k L () A jj H (n( ;h [ ;h )): Thus, exploiting hyperbolic regularity on ;h [ ;h and its complement, we deduce that jn (u) N (u h )j c 5 c 6 0 +c 5 c 6 0 c 5 c 6 c 7 0 +c 5 c ;h [ ;h khr h k L () ;h [ ;h kr h k L () A ;h [ ;h kh = r h k L () 6 ;h [ ;h khr h k L () A A k k H (n( ;h [ ;h )) A k k L() k k H (); k k L ( ;h [ ;h ) and hence the desired result. Now choosing =, the characteristic function of, we deduce from Theorem 4. that j Z (a ^)u ds Z (a ^)u h dsj c 5 c 6 c 7 0 +c 5 c 6 jj = 0 ;h [ ;h kh = r h k L () A 6 ;h [ ;h khr h k L () where jj denotes the one-dimensional measure of the set. This error bound indicates that in order to reduce the size of the error in the normal ux across one has to rene only those elements that have non-empty intersection with the domain of inuence for. Note also, that since, at least on uniformly regular partitions, the number of entries in the two sums on the right are, respectively, O(h ) and O(h ), the two terms on the right-hand side are commensurable. Due to theoretical results about nite speed of propagation of singularities along bi-characteristics and the microlocal regularity theorem of Hormander, one would expect that a bound akin to Theorem 4. holds more generally for a large class of hyperbolic systems. A ;

19 9 5 Unsteady problems In this section we develop the a posteriori error analysis of a class of evolution- Galerkin nite element methods for unsteady scalar hyperbolic problems. For a detailed overview of the (unconditional) stability and accuracy properties of evolution-galerkin methods we refer to [0]. Given a nal time T > 0, a function f L (I; L ()) with I = (0; T ], and u 0 L (), we consider the problem u t + a ru = f; x ; t I; (5.) u(x; 0) = u 0 (x); x ; (5.) where = R d. For the sake of simplicity we assume that the velocity eld a is in C([0; T ]; [C 0()] d ), that it is incompressible, i.e. ra = 0 on [0; T ], and that the supports of u 0 and f are compact subsets in and [0; T ], respectively. This problem has a unique weak solution u in L (I; L ()); moreover, if u 0 H () and f L (I; H ()) then u belongs to L (I; H ()). Let 0 = t 0 < t < : : : < t M < t M+ = T be a subdivision (not necessarily uniform) of I, with corresponding time intervals I n = (t n ; t n ] and time steps k n = t n t n. For each n, let T n = fg be a partition of into closed simplices, with corresponding mesh function h n, satisfying C y h meas() 8 T n ; (5.3) C h h n (x) h 8x 8 T n ; (5.4) where h is the diameter of, and C y and C are positive constants. Further, h will denote the global mesh function given by h(x; t) = h n (x) for (x; t) in I n, and we dene the corresponding time step function k = k(t) by k(t) = k n ; t I n. Let n = I n ; for p; q N 0, let S hn = fv H () : vj P p () 8 T n g; q V hn = fv : v(x; t)j n = j=0 t j v j ; v j S hn g; V h = fv : v(x; t)j n V hn ; n = ; : : : ; M + g; where P p () denotes the set of polynomials of degree at most p over. In the following, we shall assume that p = and q = 0. We note that if v V hn for n = ; : : : ; M +, then v is continuous in space at any time, but may be discontinuous in time at the discrete time levels t n. To account for this, we introduce the notation v n := lim s!0 + v(tn s), and [v n ] := v n + v n. The evolution-galerkin approximation of (5.), (5.) makes use of the particle trajectories (or characteristics) associated with equation (5.): the path (x; s; )

20 0 of a particle located at position x at time s [0; T ] is dened as the solution of the initial value problem d (x; s; t) dt = a((x; s; t); t); (5.5) (x; s; s) = x: (5.6) For u smooth enough, the material derivative D t u is then dened by D t u(x; s) := d dt u((x; s; t); t) j t=s = u(x; s) + a(x; s) ru(x; s) 8x ; s I: (5.7) t The evolution-galerkin time-discretisation involves approximating the material derivative by a divided dierence operator. The simplest appropriate discretisation is the backward Euler method, giving, for n = 0; : : : ; M, D t u(; t n+ ) u(; tn+ ) u((; t n+ ; t n ); t n ) k n+ : If we now let u n h denote the Galerkin nite element approximation to u(; t n ) at time t n, then applying the nite element method in space yields what is known as the evolution-galerkin discretisation of (5.): nd u n+ h S h n+, for n = 0; :::; M, such that! u n+ h u n h((; t n+ ; t n )) ; v = ( f; v) 8v S h n+ ; (5.8) k n+ (u 0 h ; v) = (u 0; v) 8v S h 0 ; (5.9) where fj n+ := f(; t n+ ). Alternatively, by integrating (5.8) with respect to t over I n+, we obtain the following equivalent formulation: nd u h such that, for n = 0; ; : : : ; M, u h j n+ V h n+ and satises where (D h t u h; v) n+ = ( f; v) n+ 8v V h n+ ; (5.0) (u 0 h ; v) = (u 0 ; v) 8v V h 0 ; (5.) D h t u h j n+ = (u h ((x; t n+ ; t n+ ); t n+ ) u h ((x; t n+ ; t n ); t n ))=k n+ ; here, for v; w L (I n+ ; L ()), we have used the notation (v; w) n+ = Z t n+ t n (v; w)dt: We remark that in (5.0), (5.) the space discretisation may vary in both space and time, but the time steps are only variable in time and not in space. Hence, the corresponding mesh will not be fully optimal.

21 5. A posteriori error analysis In this subsection we derive an a posteriori estimate for the error, e = u u h, in the L (0; T ; H ()) norm, where u and u h are the solutions of (5.), (5.) and (5.0), (5.), respectively. We shall rst introduce the following notation: for v and w in L (0; T ; L ()) and Q := I, we dene (v; w) Q := M n=0 Z t n+ t n kvk Q := ((v; v) Q ) = ; (v; w)dt; where (; ) is the inner product in L (); throughout this section k k denotes the L () norm. 5.. Error representation Given that C 0 (), consider the dual problem: nd such that t r (a) = 0; x ; t [0; T ); (5.) (x; T ) = (x); x : (5.3) Multiplying (5.) by e and integrating by parts in both space and time gives (e(t ); ) = (e(t ); (T )) = = Z T 0 M n=0 (e; t )dt + Z t n+ t n M n=0 Z t n+ t n (e t ; )dt M (f a ru h ; )dt M n=0 n=0 If we now let V h, then using (5.0) we have (e(t ); ) = M n=0 + + Z t n+ M t n n=0 Z t n+ M n=0 Z t n+ t n ([u n h ]; (tn )) + (u 0 u 0 h ; (0)) ([u n h]; (t n )) + (u 0 u 0 h ; (0)): ([u n h]=k n+ + a ru h f; )dt (D h t u h ([u n h]=k n+ + a ru h ); )dt t n ([u n h]=k n+ ; (t n ))dt + M n=0 Z t n+ t n (f f; )dt +(u 0 u 0 h ; (0)) I + II + III + IV + V 8 V h : (5.4) It remains to estimate each of the terms I to V.

22 5.. Interpolation/projection estimates for the dual problem We shall now choose V h in (5.4) as a suitable approximation to. To do so, consider the quasi-interpolation operator P n : L ()! S hn in the spatial variables (see, for example, Bernardi [3]), and the L orthogonal projector n : L (I n )! P 0 (I n ) in time dened by Z t n ( n )vdt t n = 0 8v P 0 (I n ); (5.5) then, we dene (locally) j n V hn by letting j n = P n n = n P n V hn ; where = j n. Further, we introduce P and by and dene V h as (P )j n = P n (j n); (5.6) ()j n = n (j n); (5.7) = P = P V h : (5.8) It follows from the properties of the quasi-interpolant that kk Q C i kk Q ; (5.9) with C i a positive constant, and that the following theorem holds. Theorem 5. Let n N 0 and suppose that T n satises conditions (5.3), (5.4). Let w H () have compact support; then, with C i > 0 a constant, kh n (P n w w)k L ()! = C i C jwj H (); (5.0) kp n wk L ()! = C i kwk: (5.) In the next two lemmas we provide bounds for the operators P and, in order to estimate = P. Lemma 5. Suppose that R L (I; L ()); then j(r; P ) Q j C i C khrk Q krk Q : (5.)

23 3 Proof Using the Cauchy-Schwarz inequality gives j(r; P ) Q j khrk Q kh (P )k Q : Now and kh (P )k Q = M n=0 Z t n+ t n kh n+ (P n+ )k! = ; (5.3) Therefore n+ (P n+ )k C i C krk: kh kh (P )k Q C i C krk Q : The proof of the next Lemma is given in [8]. Lemma 5.3 Suppose that R L (I; L ()); then j M n=0 j(r; P ( )) Q j C i Ci 3kkRk Q k t k Q ; (5.4) Z t n+ where C i 3 = = p and n = (x; t n ). t n (R; n )dtj kkrk Q k t k Q ; (5.5) 5..3 Strong stability of the dual problem This brief subsection summarises the stability estimates for the solution of the dual problem (5.), (5.3) in terms of the corresponding nal datum. Lemma 5.4 There exists a constant C s such that kk L(I;L ()) C s k k: Lemma 5.5 There exists a constant C s = Cs (T; a) such that krk Q C s kr k: Lemma 5.6 There exists a constant C s = 3 Cs 3 (T; a) such that k t k Q C s 3k k H (): Recalling that ra = 0, a straightforward energy analysis shows that C s =, C s = max (; e T ) where is any real number such that ra I, and C s 3 = C s kak L().

24 Completion of the proof of the a posteriori error estimate We shall now proceed to estimate the terms I-V on the right-hand side of (5.4). For the rst term I, we have I = M n=0 Z t n+ t n ([u n h]=k n+ + a ru h f; )dt = (R ; ) Q = (R ; P ) Q + (R ; P ( )) Q I + I ; where P and are as dened by (5.6) and (5.7), and By Lemma 5. we have ji j C i C Similarly, using Lemma 5.3, we have Hence, where where R j n+ = [u n h]=k n+ + a ru h f: khr k Q krk Q CC i C s khr k Q k k H (): ji j C i C i 3kkR k Q k t k Q C i C s 3C i 3kkR k Q k k H (): jij CC i C s khr k Q + CC i 3C s 3kkR i k Q k kh (): Next we consider term II: II kkr k Q kk Q C i kkr k Q kk Q C i p TkkR k Q kk L(I;L ()) C s Ci p TkkR k Q k k; (5.6) R j n+ = (D h t u h ([u n h ]=k n+ + a ru h ))=k n+ : For the term III, using Lemma 5.3 and Lemma 5.6, we have Now consider the term IV: jiiij kkr 3 k Q k t k Q C s 3kkR 3 k Q k k H (); R 3 j n+ = [u n h ]=k n+ = (u n+ h u n h )=k n+: jivj kf fk Q kk Q C i kf fk Q kk Q C i p Tkf fkq kk L(I;L ()) C s C i p Tkf fkq k k:

25 5 Finally, for term V, the Cauchy-Schwarz inequality and Lemma 5.4 give the following chain of inequalities: jvj ku 0 u 0 h k k(0)k ku 0 u 0 h k kk L(I;L ()) C s ku 0 u 0 h k k k: Substituting the bounds for the terms I to V into (5.4), dividing both sides by k k H () and taking the supremum over all in C 0 (), we obtain the following a posteriori error bound. Theorem 5.7 Let u and u h be solutions of (5.), (5.) and (5.0), (5.), respectively. Then ku u h k L(0;T ;H ()) E(u h ; h; k; f); (5.7) where E(u h ; h; k; f) = E(u h ; h; k; f) + E 0 (u 0 ; u 0 h ; h); E(u h ; h; k; f) = C khr k Q + C kkr k Q +C 3 kkr k Q + C 4 kkr 3 k Q + C 5 kkr 4 k Q ; (5.8) E 0 (u 0 ; u 0 h ; h) = C 6 ku 0 u 0 h k; (5.9) and R j n+ = [u n h ]=k n+ + a ru h f; R j n+ = (D h t u h ([u n h]=k n+ + a ru h ))=k n+ ; R 3 j n+ = [u n h ]=k n+; R 4 = (f f)=k; and C i, i = ; :::; 6, are (computable) positive constants; namely, C = C i Cs C, C = C i C s 3C i 3, C 3 = C s C i p T, C4 = C s 3, C 5 = C s C i p T, and C6 = C s. 5. Computational implementation In Theorem 5.7 we stated an a posteriori error estimate in the L (H ) norm, for the evolution-galerkin approximation of the unsteady hyperbolic problem (5.), (5.), of the form: ku u h k L(I;H ()) E(u h ; h; k; f); (5.30) where E(u h ; h; k; f) is a computable global error estimator. We shall now consider the problem of implementing the error bound (5.30) into an adaptive evolution-galerkin nite element algorithm to ensure global

26 6 error control (and thereby reliability) with respect to a prescribed tolerance, TOL, i.e. ku u h k L(I;H ()) TOL; (5.3) with the minimum amount of computational eort. We begin, in Subsection 5.. by designing an adaptive algorithm to determine the mesh parameters h and k in such a way that (5.3) holds. Then, in Subsection 5.., we describe how the space-time mesh may be locally adapted so that the actual mesh parameters h and k are `close' to those predicted by the adaptive algorithm. 5.. Adaptive algorithm For a given tolerance, TOL, we consider the problem of nding a discretisation in space and time S h = f(t n ; t n )g n0 such that:. ku u h k L(I;H ()) TOL;. S h is optimal in the sense that the number of degrees of freedom is minimal. In order to satisfy these criteria we shall use the a posteriori error estimate (5.30) to choose S h such that:. E(u h ; h; k; f) TOL;. The number of degrees of freedom in S h is minimal. The term E 0 (u 0 ; u 0 h ; h) is easily controlled at the start of a computation; so here we shall only consider the problem of constructing S h in an ecient way to ensure that E(u h ; h; k; f) TOL 0 ; where TOL = TOL 0 + E 0 (u 0 ; u 0 h ; h). To do so, we rst write E symbolically in terms of two residual terms: one that controls the spatial mesh and one that controls the temporal mesh, i.e. we let E(u h ; h; k; f) C 0 khr 0 k Q + C 0 kkr 0 k Q : (5.3) Simultaneously, we split the tolerance TOL 0 into a spatial part, TOL h, and a temporal part, TOL k. Thus, for reliability we require that the following conditions hold: C 0 khr 0 k Q TOL h ; (5.33) C 0 kkr 0 k Q TOL k : (5.34)

27 7 To design the space-time mesh S h, at each time level t n we decompose the norm in (5.33) into norms over elements T n, and the norm in (5.34) into norms over time slabs as follows: C 0 khr 0 k Q C 0 p T max nm+ kh nr 0 (u n h)k C 0 p NT max nm+ C 0 kkr 0 k Q C 0 p T max nm+ kk nr 0 (u n h)k; max T n kh n R 0 (u n h)k L () where N is the number of elements in the spatial mesh at time t n. Thus, if C 0 p NTkhn R 0 (u n h)k L () TOL h 8 T n ; for n = ; : : : ; M + ; C 0 p Tkkn R 0 (u n h)k TOL k ; for n = ; : : : ; M + ; then (5.33) and (5.34) will automatically hold. For the practical implementation of this method, we consider the following adaptive algorithm for constructing S h, under the assumption that the nal time T is xed: for each n = ; ; : : : ; M +, with T n 0 a given initial mesh and k n;0 an initial time step, determine meshes Tj n with N j elements of size h n;j (x) and time steps k n;j and the corresponding approximate solution u h;j dened on Ij n such that, for j = 0; ; : : : ; ^n, ; C kh n;j+ R (u n )k h;j L () = TOL h q Nj T 8 T n j ; (5.35) C kk n;j+ R (u n h;j)k + C 3 kk n;j+ R (u n h;j)k +C 4 kk n;j+ R 3 (u n h;j)k + C 5 kk n;j+ R 4 (u n h;j)k = TOL k p ; (5.36) T where I n j = (t n ; t n + k n;j ] and TOL 0 = TOL h + TOL k. We dene T n = T n^n, k n = k n;^n and h n = h n;^n, where for each n, the number of trials ^n is the smallest integer such that for j = ^n, the following stopping condition is satised: C kh n;^n R (u n h;^n)k L () TOL h p N^n T C kk n;^n R (u n h;^n)k + C 3 kk n;^n R (u n h;^n)k 8 T n^n ; (5.37) +C 4 kk n;^n R 3 (u n h;^n)k + C 5 kk n;^n R 4 (u n h;^n)k TOL k p : (5.38) T By construction, this stopping condition will guarantee reliability of the adaptive algorithm; for eciency, we try to ensure that (5.37) and (5.38) are satised with near equality. Since the nal time T is xed, the time step given by (5.36) may need to be limited to ensure that t M + k M+;^n = T. For the implementation of this adaptive algorithm, we shall assume that T n 0 = T n for n = ; 3; : : :.

28 8 5.. Mesh modication strategy In Subsection 5.. we described the construction of the space-time mesh S h to achieve the required error control, ku u h k L(I;H ()) TOL: However, to obtain the desired mesh we must employ a suitable mesh modication technique. Temporal adaptation is quite straightforward, since the time step can just be set equal to k n;j+ given by (5.36). For constructing the spatial mesh in two space dimensions we use the red-green isotropic renement strategy of Bank [4]. Here, the user must rst specify a (coarse) background mesh upon which any future renement will be based. Red renement corresponds to dividing a certain triangle (father) into four similar triangles (sons) by connecting the midpoints of the sides. Green renement is only temporary and is used to remove any hanging nodes caused by a red renement. We note that green re- nement is only used on elements which have one hanging node. For elements with two or more hanging nodes a red renement is performed. The advantage of this renement strategy is that the degradation of the `quality' of the mesh is limited since red renement is obviously harmless and green triangles can never be further rened. Within this mesh modication strategy it is also possible to de-rene the mesh by removing redundant elements, provided that these do not belong to the original background mesh. Thus, to prevent an overly rened mesh in regions where the solution is smooth the background mesh should be chosen suitably coarse. For the practical implementation of this mesh modication strategy we have used the FEMLAB package developed by K. Eriksson. 5.3 Numerical experiments In this section, we present some numerical experiments to illustrate the performance of the adaptive algorithm (5.35), (5.36) on the model hyperbolic test problem: u t + a ru = f; x ; t I; (5.39) u(x; 0) = u 0 (x); x ; (5.40) where = (0; ), f = 0, a = (; ) T, subject to the boundary condition u(0; y) = for 0 y, u(x; 0) = ( x) + = for 0 x. The function u 0 appearing in the initial condition is dened as follows: u 0 (x) = 0 for x = (; ) (0; ); and for x n, u 0 (x) is chosen to be the linear function that satises the boundary conditions at inow. We note that initially, for small, the solution to this problem has a boundary layer along x = 0; this layer then propagates into

29 9 (a) (b) Figure : (a) Background mesh, with 5 nodes and 3 elements; (b) Background mesh adapted to resolve the initial condition, with 56 nodes and 900 elements. the domain, and eventually exits through x =. In the following, we shall let = 7: and T = 0:55. First, we specify the background mesh to be the 55 triangular mesh shown in Figure (a). This mesh is initially rened in order to properly resolve the boundary layer along x = 0 at time t = 0, i.e. so that E 0 (u 0 ; u 0 h ; h) = 0 (see Figure (b)). Numerical results are presented in Figure for TOL h = 0:0 and TOL k = 0:5. The estimated error is E(u h ; h; k; f) = : In Figures (a), (b) and (c), (d) we see that the adaptive algorithm has rened the spatial mesh in parts of the domain where the solution has a steep layer, and has kept the mesh coarse elsewhere. Figures (e), (f) show the history of the number of nodes in the spatial mesh against time, and the size of the time step against time, respectively. We note that the articial diusion model introduced in [9] was used in the above experiment with C ^ = C ^ = 0: and ^ max = 5: Conclusions In this paper we have presented an overview of recent developments that concern the a posteriori error analysis of nite element approximations to rst-order hyperbolic problems. While for elliptic equations there is a well-established theoretical framework of a posteriori error estimation that has been successfully implemented into working adaptive algorithms (see, for example, the monograph of Ainsworth and Oden [] for an extensive list of literature in this area), very much less is known about hyperbolic and nearly-hyperbolic problems. The main problem both from the theoretical and the practical point of view is the estimation of the stability constant of the dual problem that enters the nal error