Introduction The numerical solution of rst-order hyperbolic problems by nite element methods has become increasingly popular in recent years. Two majo

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1 Report no. 98/4 Stabilized hp-finite Element Methods for First{Order Hyperbolic Problems Paul Houston Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford O 3QD, U Ch. Schwab Seminar of Applied Mathematics, ETH urich, CH-809 urich, Switzerland Endre Suli Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford O 3QD, U We analyze the hp-version of the streamline-diusion (SDFEM) and of the discontinuous Galerkin method (DGFEM) for rst{order linear hyperbolic problems. For both methods, we derive new error estimates on quadrilateral meshes which are sharp in the mesh-width h and in the spectral order p of the method, assuming that the stabilization parameter is O(h=p). For piecewise analytic solutions, exponential convergence is established. For the DGFEM we admit very general irregular meshes and for the SDFEM we allow meshes which contain hanging nodes. Numerical experiments conrm the theoretical results. Oxford University Computing Laboratory Numerical Analysis Group Wolfson Building Parks Road Oxford, England O 3QD November, 998

2 Introduction The numerical solution of rst-order hyperbolic problems by nite element methods has become increasingly popular in recent years. Two major families of methods have emerged: the streamline diusion method (SDFEM) and the discontinuous Galerkin method (DGFEM). While the SDFEM uses continuous approximations, the DGFEM allows for discontinuities at element interfaces and is, in spirit, close to the well-established nite volume schemes with some particular dierences, however. For classical nite element and nite volume methods improvement in accuracy relies on mesh renement while keeping the approximation order within the elements (or cells) at a xed, low value, leading to the so-called h-version convergence. In the late seventies and early eighties, however, the so-called p-version or spectral methods emerged which achieve convergence by increasing the polynomial order of the approximation rather than by mesh renement. Naturally, this is very advantageous in situations where a smooth or even analytic solution is to be approximated. Unfortunately, the solution to most problems of practical interest is only piecewise analytic: in elliptic problems (such as stationary viscous incompressible ow), corner and edge singularities arise in the vicinity of which the solution regularity is very low. Good performance of high order methods and, in particular, spectral or exponential convergence for such problems mandates the combination of increasing polynomial degree in regions where the solution is smooth and mesh renement with low order polynomial approximations close to singularities. This strategy gives rise to the hp-version of the nite element method introduced by Babuska and his co-workers in the mid-eighties for elliptic problems. The DGFEM has been proposed and rst analyzed in [9] for a linear hyperbolic problem. There, the method was formulated and its h-version convergence was established in L (), albeit with a suboptimal rate. Later, in [7], [8], the optimal rate of O(h p= ) in a meshdependent norm (stronger than L ()) was proved, assuming that the nite element space consisted of piecewise polynomials of degree p. In the meantime, the DGFEM has also been successfully applied to nonlinear hyperbolic conservation laws (see, e.g., [4]). The hp-version of the DGFEM has been introduced by. Bey and J.T. Oden, who gave a-priori and a-posteriori error bounds in []. Their analysis produced error estimates which, for a xed p and as h! 0, reduced to the optimal order estimates of [7], [8], but also indicated convergence as p! for xed h > 0. These results were derived under the assumption that the stabilization parameter in element is of size h =p ; however, the rate of this spectral convergence was suboptimal. In the present paper, we generalize the results of [] in several directions. We establish a unied framework for the hp-error analysis of the SDFEM and the stabilized DGFEM; on quadrilateral meshes we derive error estimates which are sharp both as h! 0 and as p!. These optimal error bounds are derived assuming that the stabilization parameter for both the DGFEM and the SDFEM, and for the h-, p- and the hp-version is O(h =p ), independent of the solution regularity. For the DGFEM we admit very general, irregular meshes and for the SDFEM we allow meshes which contain hanging nodes. Most importantly, our error estimates depend explicitly on the elemental solution regularity and indeed allow us to deduce exponential convergence rates for piecewise analytic solutions. The theoretical ndings are in full agreement with the numerical experiments which complete the

3 3 paper. We note in closing that Bey and Oden [] also considered the a-posteriori error analysis of the hp-dgfem. Using the sharp error estimates obtained here, new a-posteriori error estimates can be derived for hp-dgfem and hp-sdfem. However, this subject is beyond the scope of the present paper and will be considered elsewhere. The model problem Let be a bounded curved polyhedral domain in lr d, d. Given that a = (a ; : : : ; a d ) is a d-component vector function dened on with a i C (), i = ; : : : ; d, we dene the following subsets of = fx : a(x) n(x) < 0g ; = fx : a(x) n(x) > 0g ; where n(x) denotes the unit outward normal vector to at x. It is assumed here implicitly that in these denitions x ranges only through those points of at which n(x) is dened; consequently, and are not necessarily connected subsets of. For the sake of simplicity, we shall suppose that is non-characteristic in the sense that [ =. Let b C(), f L (), g L ( ) and consider the hyperbolic boundary value problem ( Lu a ru bu = f in ; u = g on : (.) This problem has a unique weak solution u L () with a ru L () and the boundary condition is satised as an equality in L ( ). In the next two subsections we shall formulate the hp-streamline diusion and hpdiscontinuous nite element approximation of (.).. hp-finite Element Spaces.. Meshes Let P denote a partition of into open patches P which are images of a reference domain ^P under smooth, bijective maps F P : 8P P : P = F P ( ^P ) : We assume that ^P is either the canonical cube ^P = ^Q := ( ; ) d or the unit simplex ^P = ^S := n ^x lr d : ^x i > 0; d o ^x i < : i=

4 4 ( ; ) (; ) ^ F P ( ; ) (; ) Patch ^P and Mesh ^T P Patch P and Mesh T P Figure : Construction of the mesh patch T P in the case when ^P is the canonical cube ^Q. At this stage, we consider meshes which result from either ^Q or ^S; in Section 3 and onwards, for the sake of simplicity, we shall restrict ourselves to the case of d = and quadrilateral meshes. The meshes T are constructed by subdividing the patches. For each P, a mesh T P is obtained by rst subdividing ^P into elements (e.g. triangles resp. quadrilaterals when d = ) labelled ^ which are ane equivalent to either ^Q or ^S; we call this mesh ^T P. A mesh T P for P P is then obtained by simply mapping ^T P to P using F P : 8P P : T P := f j = F P ( ^); ^ ^TP g ; (.) cf. Figure. As usual, the mesh T in is the collection of all elements, i.e. [ T = T P : P P Note that each T is an image of the reference domain ^P via the element map F : if P for some P P, and A ^ : ^P! ^ ^T P is ane. = F ( ^P ); F := F P A ^ (.3) Remark. The maps F P, P P, are assumed to only deform the canonical patch ^P without any signicant rescaling, thereby ensuring that the measure of the set ^P is comparable to the measure of set P ; thus we may infer that the elements ^ in the mesh ^TP are of

5 5 comparable size to the elements in the mesh T P. More explicitly, we assume there exist positive constants c and c such that for all in the mesh T c h =h ^ c (.4) with h = diam(), h ^ = diam( ^) and ^ is associated with via = FP ( ^), as in (.). This will be important as our error estimates will be expressed in terms of Sobolev norms over the element domains ^, in order to ensure that only the scaling introduced by the ane element maps A ^ is present in the analysis. We emphasize that we could choose A ^ and F P in (.3) so as to obtain the usual parametric elements. However, it is also possible to use patches P with structured patch-meshes T P, as e.g. geometric corner renement, anisotropic boundary layer and edge renement etc. In what follows, the partition P shall be xed, i.e. mesh renement is performed in ^P. We call the mesh T regular, if for any two ; 0 T the intersection \ 0 is either empty or an entire boundary segment of dimension 0 d 0 < d (e.g. a vertex (d 0 = 0), an entire edge (d 0 = ), an entire side (d 0 = ) etc.). If the mesh T is regular, the maps F P are assumed compatible between patches in the sense that if P \ P 0 6= ; : F P j P \P 0 = F P 0j P \P 0; i.e. F P (x) = F P 0(x) 8x P \ P 0 : (.5) The T P are -irregular, if they consist of quadratics resp. hexagonal elements with at most one irregular (\hanging") node per side. T is -irregular, if the T P T are either regular or -irregular and compatible between patches... Polynomial spaces On the reference element we dene spaces of polynomials of degree p 0 as follows: Q p = spanf^x : 0 i p; i dg; P p = spanf^x : 0 jj pg: (.6)..3 Polynomial subspaces on ^P Let T be any mesh as in.. and let p = fp : T g be a polynomial degree vector on T. The denition of a discontinuous hp-fe space is now straightforward: if F P = ff P : P P) denotes the patch-map vector, we set S p;0 (; T ; F P ) := fu L ()j uj F Q p if T is quadrilateral resp. uj F P p if is triangularg : (.7) No inter-element continuity is imposed here. If the polynomial degree is uniform, p = p for all T, we write S p;0 (; T ; F P ). If the choice of, T and F P is clear from the context, we omit them and write S p;0.

6 6 Let us now turn to continuous hp-fe spaces. Here we assume T to be either regular or -irregular. If the polynomial degrees p are uniform, namely p = p for all, we dene, for p, S p; (; T ; F P ) = S p;0 (; T ; F P ) \ H () ; (.8) i.e. inter-element continuity is now enforced and the compatibility condition (.5) between patches is required. If the polynomial degrees are nonuniform, there are several ways to enforce inter-element continuity - assume that ; 0 T share a d dimensional set, and that p < p 0. One can now either enrich the polynomials on or constrain the polynomials on 0. We adopt here the latter approach and set S p; (; T ; F P ) = S p;0 (; T ; F P ) \ H () : (.9) Note that one could even allow anisotropic/nonuniform polynomial degrees within an element T - this becomes important when adaptivity is considered (see [5] and the references therein). Denition (.9) implies that the degrees of freedom from 0 that are unmatched by those from are constrained to zero on interfaces \ 0.. The hp-sdfem The hp-sdfem approximation of (.) is dened as follows: nd u SD S p; such that (Lu SD ; v Lv) (u SD ; v) = (f; v Lv) (g; v) 8v S p; ; (.0) where is a positive piecewise constant function dened on the partition T (namely, is constant on each T ). In (.0), (; ) denotes the inner product of L (), and (w; v) = ja nj wv ds ; with analogous denition of (; ) and associated norms k k and k k. Our rst result concerns the stability of the hp-sdfem and is expressed in the next lemma. Lemma. Suppose that there exists a positive constant c 0 such that b(x) r a(x) c 0; x : (.) Then u SD obeys the bound k p Lu SD k c 0 ku SD k ku SD k ku SDk k p fk c 0 kfk kgk : (.)

7 7 Proof: Select v = u SD in (.0) and note that (Lu SD ; u SD ) (u SD ; u SD ) = b r a u SD ; u SD ku SDk ku SDk : (.3) Applying (.) here and using the Cauchy{Schwarz inequality on the right-hand side in (.0) with v = u SD, the result follows. Now we embark on the error analysis of (.0). We begin by decomposing u u SD = (u u) (u u SD ) ; (.4) where u is a suitable projection of u into S p; ; for the time being the choice of the projector is of no signicance and will be deferred until later. First we shall derive a bound on in terms of ; the nal error bound on u u SD will then follow from bounds on the projection error. Lemma.3 Assuming that (.) holds, and u H (), we have that k p Lk kck kk kk k p L p where c C() is dened by k 4kck kk ; (.5) c (x) = b(x) r a(x); x : (.6) Proof: Dene the bilinear form B(w; v) = (Lw; v Lv) (w; v) (.7) for w; v H () and the linear form `(v) = (f; v Lv) (g; v) ; (.8) for v H (). Then, from (.4), B(; ) = B(u u SD ; ) = B(u; ) B(u SD ; ) B(; ) : (.9) Since u and u SD solve (.) and (.0) respectively, it follows that From (.9) and (.0) we have, B(u; ) B(u SD ; ) = B(u; ) `() = 0 : (.0)

8 8 B(; ) = B(; ) : (.) Applying (.3) from the proof of Lemma. with u SD replaced by, k p Lk kc k kk kk = B(; ) : (.) The rest of the proof is devoted to bounding B(; ). By partial integration, Hence B(; ) = p B(; ) p L; p p L (c ; ) (; ) : p L kp Lk kc k kc k 4 kk kk : (.3) Substituting (.3) into (.) and multiplying the resulting inequality by gives (.5)..3 The hp-dgfem Given that is an element in the partition T, we denote the union of open faces of. This is non-standard notation in is a subset of the boundary of. Let and suppose that n(x) denotes the unit outward normal vector at x. With these conventions, we dene the inow and outow parts respectively, = : a(x) n(x) < 0g = : a(x) n(x) 0g : For each T and any v H () we denote by v the interior trace of v (the trace taken from within ). Now consider an element such that the n is nonempty; then for each n (with the exception of a set of (d ) dimensional measure zero) there exists a unique element 0, depending on the choice of x, such that 0. This is illustrated in Figure. Now suppose that v H () for each T. n is nonempty for some T, then we can also dene the outer trace v of v n relative to as the inner trace v relative to those elements 0 for 0 has intersection n of positive (d )-dimensional measure. We also introduce the jump of v n : [ v ] = v v : Let H () for each T, and suppose that is positive on each T. Typically, is chosen to be constant on each T, although we shall not require this for now.

9 9 0 x * a Figure : A point x such that and 0. Suppose that v; w H () for each T. We dene B DG (w; v) = Lw (v Lv) dx (a n)[w] v n (a n)w v \ (.4) and put `DG (v) = f (v Lv)dx (a n) gv ds : \ The hp-dgfem approximation of (.) is dened as follows: nd u DG S p;0 such that B DG (u DG ; v) = `DG (v) 8v S p;0 : (.6) Next we study the stability of the discrete problem (.6). Lemma.4 Suppose that there exists a positive constant c 0 such that (.) holds. Then u DG obeys the bound k p Lu DG k c 0 ku DG k ku DG \ k p fk c0 kfk ku DG u DG n ku DG \ \ : (.7)

10 0 Remark: This bound is analogous to the estimate (.) for the hp-sdfem. Proof: Take v = u DG in (.6); this gives B DG (u DG ; u DG ) = `DG (u DG ): (.8) We begin by bounding the left hand side in (.8) from below. Upon partial integration, (.4) gives B DG (u DG ; u DG ) = jlu DG j dx b r a ju DG j (a n)ju DG j n (a n)[u DG] \ (a n)ju Now into the union of four disjoint sets and writing DG ds DG j ds = (@ n ) [ (@ \ ) [ (@ n ) [ (@ \ ) [u DG ] u DG = (u DG u DG ) u DG = ju DG j (u DG u DG ) ju DG j ; (.9) the last three terms in (.9) can be rewritten n (a n)judg j ds (a n)ju DG u DG j \ (a n)ju DG j ds : (.30) Here we made use of the fact that (a n)judg j n (a n)judg j ds = 0 n

11 Using (.30) in (.9) yields B DG (u DG ; u DG ) k p Lu DG k c 0 ku DG k ku DG \ ku DG u DG n ku DG \ : (.3) Now we bound the right-hand side in (.8) using (.5): j`dg (u DG )j kfk ku DG k k p fk k p Lu DG k c 0 4 \ ku DG \ ku DG k c 0 k p Lu DG k ku DG \ kfk k p fk \ : (.3) Inserting (.3) and (.3) into (.8) gives (.7). We now discuss the error analysis of hp-dgfem. We write u u DG = (u u) (u u DG ) ; (.33) where u is a suitable projection of u into S p;0, to be chosen below. Lemma.5 Assuming that (.) holds and u H () for each T. We have that k p Lk kc k k \ k p L p k 4 k \ k n kc k (.34) k \ k n :

12 Proof As in Lemma.3, B DG (; ) = B DG (; ) : Applying (.3) with u DG replaced by gives k p Lk Next we transform B DG (; ): Now B DG (; ) = kc k k \ L L dx b r a dx L n (a n (a n) 4 Substituting (.37) into (.36), we get B DG (; ) k \ k n jb DG (; )j : (a n) \ ds ds k n k n k n 4 4 k p L k kc k (a n) n (a n) n k \ k n : kc k k n k p L p k k \ k n : (.35) (.36) (.37) (.38) Now inserting (.38) into (.35) gives (.34).

13 3 3 hp-error Estimates In this section, we shall construct the hp-approximation projector in the error estimates (.4), (.33) and derive hp-error bounds for the hp-sdfem as well as for the hp-dgfem introduced in the previous section. The bounds are explicit in h and p and in the regularities of the solution and allow us to deduce in particular exponential convergence estimates for piecewise analytic solutions. For simplicity, we restrict ourselves to d = space dimensions and to meshes consisting of quadrilateral elements. 3. One-dimensional hp-approximation We cite some approximation results from [0]. To this end, we set ^I = ( ; ) and denote by kuk k; ^I resp. juj k; ^I the Hk (^I) norm resp. seminorm on ^I. Denote further S p (^I) the polynomials of degree p on ^I. Then we have Theorem 3.6 Let u H k (^I) for some k 0. p u S p (^I) such that for any 0 s min(p; k) and such that Then, for every p, there exists ku 0 ( p u) 0 (p s)! k ^I (p s)! juj s; ^I (3.) ku p uk ^I p(p ) for any 0 t min(p; k). Moreover, we have For the proof, we refer e.g. to [0]. (p t)! (p t)! juj t; ^I (3.) p u( ) = u( ) : (3.3) Corollary 3.6A The projector p whose existence is asserted in Theorem 3.6 is bounded as follows: k( p u) 0 k ^I ku 0 k ^I ; (3.4) k p uk ^I kuk ^I pp(p ) ku0 k ^I (3.5) for all p and every u H (^I). Proof: The inequality (3.) with s = 0 implies k( p u) 0 k ^I k( p u) 0 u 0 k ^I ku 0 k ^I ku 0 k ^I : Similarly, (3.) with t = 0 implies k p uk ^I k p u uk ^I kuk ^I kuk ^I pp(p ) ku0 k ^I :

14 4 ^ 3 ^ 4 ^ ^ Figure 3: ^Q and the notation for the sides. 3. Approximation on quadrilaterals Higher dimensional approximation results will be obtained from Theorem 3.6 by tensor product construction. We denote by p i u the one-dimensional projector in Theorem 3.6 applied to u as function of the ith coordinate alone and perform the error analysis for d =. Let ^Q = ( ; ) and denote by ^ i, i = ; ; 3; 4, the sides of ^Q as shown in Figure 3. Theorem 3.7 (Reference Element Approximation) Let ^Q = ( ; ), as in Figure 3, p and assume that u H k ( ^Q) for some k. Let p = p p denote the tensor product projector. Then there holds: p u = u at the vertices of ^Q ; (3.6) p uj^i = ( p(uj^i ) if i is odd ; p(uj^i ) if i is even : (3.7) The following error estimates hold: (p s)! kr(u p u)k ^Q (p s)! 8 p(p ) n o k@ s uk k@s ^Q uk ^Q (p s )! (p s )! n k@ uk ^Q s uk^q o ; (3.8) n o k@ s uk k@s ^Q uk ^Q ku p uk ^Q (p s)! p(p ) (p s)! 4 (p s )! p (p ) (p s )! s ; (3.9) uk^q

15 5 for any 0 s min(p; k). Proof: We prove (3.9). Clearly, ku p uk ^Q ku p uk^q k p(u p u)k^q : For the rst term we use the bound (3.), resulting in ku p uk^q p(p ) For the second term, (3.5) and (3.) give k p(u p u)k^q ku p uk^q p(p ) (p s)! (p s)! k@s uk ^Q : (p t)! (p t)! k@t uk ^Q p (p ) p(p ) k@ (u p u)k^q (p r)! (p r)! r uk ^Q : Selecting t = s and r = s gives (3.9). The proof of (3.8) is analogous. 3.3 Approximation on quadrilateral meshes with hanging nodes Consider now a mesh patch P P with mesh T P and corresponding reference mesh ^T P in ^P. We assume that all T P are quadrilateral, possibly with hanging nodes. With we associate the edge-lengths of the sides of ^ = FP () denoted by h i; ^, i = ;. Theorem 3.8 (Discontinuous Approximation) Let P P with quadrilateral, possibly -irregular mesh T P of shape-regular elements and polynomial degree distribution p. For all T P let uj H k () for some k and dene u S p;0 (P; T P ) element-wise by (u)j F P := p (uj F P ) 8 T P ; with p as in Theorem 3.7. Then, for p and for 0 s min(p ; k ) the following estimate holds: ku uk P h s C p (p ) (p ; s )j^uj ; s (3.0) ; ^ T P where ^u = u F P, = F P ( ^) and (p; s) := (p s)! (p s)! p(p ) (p s )! (p s )! ; 0 s p : (3.)

16 6 Furthermore, kr(u u)k P C T P h s(p ; s )j^uj : s (3.) ; ^ The constant C > 0 in these estimates depends only on F P, but is independent of h, p and s. Proof: The L -estimate (3.0) follows immediately by a change of variables and a scaling argument from Theorem 3.7. For the gradient estimate, we observe that kr(u u)k P C(F P )k ^r((u u) F P )k ^P : For the right{hand side we use (3.8), after scaling to the reference element: k ^r((u u) F P )k ^P = = 4 ^ ^TP ^ ^T P i=; (3:8) (h ^ ) k^@ ((u u) F P )k ^ k^@ ((u u) F P )k ^ h ; ^ h ; ^ k^@ i (I p )u F P A k ^Q ^ ^TP 4 p (p ) n (p s )! (p s )! (k^@ s u 0; k ^Q k^@ s u 0; k ^Q ) (p s )! (p s )! (k^@ s ^@ u 0; k ^Q k^@ ^@s u 0; k ^Q ) o; where u 0; := u F P A = ^u A ; T P : Ane scaling from ^Q to ^ ^T P and noting (.4) gives the assertion. The error bounds in Theorem 3.8 simplify for uniform p. Corollary 3.8A (Uniform order estimate) Assume that ^u := u F P H k ( ^P ) and that for all TP p = p ; s = s; 0 s min(p; k) : Then, for u S p;0 (P; T P ) and ^u := u F P, the following estimates hold: ku uk P C p(p ) (p; s) T P h sj^uj ; (3.3) s; ^

17 7 and kr(u u)k P C(p; s) TP h sj^uj : (3.4) s; ^ Here C > 0 is a constant that depends only on the patch mapping F P but not on s; p; h. Remark (Anisotropic error estimates) We note in passing that the above error estimate assumed the shape regularity of the ^ merely for convenience - in fact the explicit error bounds in Theorem 3.7 and 3.8 above could be easily generalized to anisotropic element shapes (with edge-lengths h and h ) and even to anisotropic polynomial degrees p, p, say. Error bounds explicit in these parameters can be deduced by inspecting the proofs of the above theorems. Theorem 3.8 addressed only discontinuous approximations; it turns out, however, that also continuous, piecewise polynomial approximations can be obtained. Theorem 3.9 (Continuous approximations) Let lr and let P P with a -irregular mesh consisting of shape regular quadrilaterals of diameter h. Let the polynomial degree be uniform, p = p. Let uj H k () for some k and let u H (P ). Then there exists a projector e u S p; (P; T P ) such that the error bounds (3.3), (3.4) hold, with a possibly dierent value of C. Proof If T P does not contain hanging nodes, T P is regular and we take e = in Theorem 3.8. Since was constructed element-wise, the properties (3.0), (3.) together with the assumption that u H (P ) give the continuity of u in P. Suppose now that T P contains hanging nodes. A typical situation in the reference mesh ^T P is shown in Figure 4 where the elements have been scaled to unit size for convenience. Since u H (P ), also u C 0 (P ). By (3.6), u u vanishes at the points in Figure 4. Denote by [u u] ij the jump of u u across ij. By (3.6), the jump of u across 3 is zero. Since u C 0 (P ), [u u] ij = [u] ij. Further, [u] ij P p ( ij ). We now construct a trace-lifting of [u] across [ 3 as follows: we set ( [u] ( ) on ^ ; V () = ( ) [u] 3 ( ) on ^3 : Since [u] 3 = 0, V is continuous on ^ [ ^3 and krv k ^ [ ^ 3 C k [u] k H ( [ 3 ) ; (3.5)

18 8 ( ; ) (; ) ^ ( ; 0) 3 (; 0) ^ ^ 3 3 ( ; ) (0; ) (; ) Figure 4: Hanging node and adjacent elements. where C is independent of p. By the trace theorem and since u C 0 ( 3 S ^ i ), we have k [u]k H ( [ 3 ) = k [u u] k H ( [ 3 ) k (u u) k H ( [ k(u 3 ) u) k H ( [ 3 ) (3.6) 3 C ku uk H ( ^ i ) ; i= where () denote traces from > 0 and < 0, respectively. We dene eu := ( u on ^ ; V u on ^ [ ^ 3 : Now u e is continuous on and across and 3. Therefore, on ^ := ^ [ ^ [ ^ 3, we have that kr(u e u)k ^ krv k ^ [ ^ 3 3 i= kr(u u)k ^i : Using (3.5), (3.6) we get where C > 0 is independent of p. kr(u e u)k ^ C 3 i= ku uk ; ^ i ; (3.7)

19 9 Now suppose that the ^ i, i = ; ; 3, are not of unit size but that their diameters are proportional to h ^, where h ^ in the diameter of ^. Performing a scaling of the independent variable by a factor of h ^ in the estimate (3.7) (noting that diam( ^ ) = diam( ^ 3 ) = h ^ =, diam( ^ ) [h ^ =; h ^ ]), recalling (.4) and inserting (3.3) and (3.4) into the resulting right-hand side, we deduce that a bound analogous to (3.4) holds with replaced by e. Concerning the analogue of the bound (3.3) with replaced by e and e dened as above, again, we consider the conguration shown in Figure 4 with the elements scaled to unit size, for convenience. Observe that kv k ^ [ ^ 3 Ck[u u]k L ( [ 3 ) C k (u u) k L ( [ 3 ) k(u u) k L ( [ 3 ) C 3 i= ku uk ^i ku uk = kr(u u)k = ; ^ i ^ i where in the transition to the last line we made use of the multiplicative trace inequality. Consequently, also ku e uk ^ C 3 i= ku uk ^i ku uk = kr(u u)k = : (3.8) ^ i ^ i Now suppose the ^ i, i = ; ; 3, are of size proportional to h ^ ; then, we may scale the independent variable by h ^ in estimate (3.8) and insert (3.3) and (3.4) into the resulting right-hand side to deduce that a bound analogous to (3.3) holds with replaced by e. Finally, we note that since ^ = ^ ; further liftings in the presence of additional hanging nodes ^ can be performed in the adjacent element patches, resulting in the error bounds (3.3), (3.4) with a larger C. 3.4 hp-error Analysis of the DG- and the SDFEM We are now in a position to present error estimates for both the SD- and the DGFEM. We shall use the following norm dened by kjukj DG := T n k p Luk kcuk ku \ ku \ ku u n o : (3.9) Notice that for the SDFEM, the last term vanishes. Here is our main error estimate for the hp-dgfem.

20 0 Theorem 3.0 (Convergence rate of the hp-dgfem) Let lr and T ; P be as in Section with (possibly irregular) mesh patches T P, P P, consisting of shape-regular quadrilateral elements of degree p. Select j = = h =p for all T : (3.0) Then kju u DG kj DG C h s (p ; s ) j^uj ; p s (3.) ; ^ where C > 0 depends only on elemental shape regularity, and the coecients a; b, but is independent of p ; s ; h and where (p; s) is as in (3.). Proof Using (.33) and Lemma.5 gives kju u DG kj DG kjkj DG kjkj DG (:34) kjkj DG kck k L k p k \ k n : Therefore, kju u DG kj DG p C k Lk k \ n kc k p k n kc k p k rk k k \ k Lk k \ k kk k k k k k n k \ k \ k n k n o C(A B) ; where C depends on (a; b). We select = u u with as in Theorem 3.8. This gives the bound h s(p A C ; s )( h p )j^uj s ; ^ :

21 To bound B, we must estimate We use the inequality and obtain the bound B C = C C (krk kk h kk ) h h h h s(p ; s ) h s (p ; s ) p j^uj s ; ^ 8 T s (p ; s ) p j^uj s ; ^ s p (p ; s )( p )j^uj s ; ^ : Selecting as in (3.0) concludes the proof. An analogous error estimate holds true for the hp-sdfem. Theorem 3. (Convergence rate of the hp-sdfem) Let lr and T ; P be as in Section with a -irregular mesh consisting of shape-regular quadrilateral elements of degree p. Select the stabilization parameter as in (3.0). where Then there holds the error estimate kju u SD kj SD C h s (p ; s ) p j^uj s ; ^ ; (3.) and kjukj SD := kp Luk kcuk kuk kuk 0 s p 8 T ; ^u = u F P if T P ; and (p; s) is as in (3.). The proof of Theorem 3. is completely analogous to that of Theorem 3.0, using Lemma.3 instead of Lemma.5. Let us now discuss some special cases of the above, general error bounds. Remark 3. ) If p = p is xed, and h = h! 0, the bound (3.) is optimal in h. ) As s is xed and p = p!, Stirling's formula implies and (3.) gives (p; s) C(s) p s kju u DG kj DG C h The bound (3.) is therefore optimal also in p. p s j^uj s ; ^ :

22 3) Suppose that u is patch-wise analytic. Then, 8 ^ T 9d > ; C > 0 8s > 0 : j^uj s; ^ C(d ) s s! (3.3) In this case, (3.) gives exponential convergence, since picking s = p, with 0 < < to be selected below, and applying Stirling's formula gives (p; s)j^uj s; ^ C(d ) s ((s )!) (p s)! (p s)! C(d ) p (p ) p3 e p (( )p)( )p e ( )p (( )p) ()p e ()p Cp 3 (F (; d )) p ; where F (; d) := ( ) ( ) (d) : Since, for d >, min F (d; ) = F (d; min) < ; min = 0<< p d ; it follows, setting b = j log F (d ; min )j, that (p ; p ) j^uj p ; ^ Cp3 e b p ; and we get from (3.) the exponential convergence estimate h kju u DG kj DG C s p e b p : By Theorem 3. an analogous bound holds also for the hp-sdfem on quadrilateral, possibly -irregular meshes. Finally, we note that exponential convergence estimates analogous to the ones presented here on quadrilaterals can also be proved on triangular meshes, using the approximation results of Braess and Schwab []. Further aspects of the local discontinuous Galerkin method will be considered in [3]. 4 Numerical experiments In this section we present a number of numerical experiments to verify the a priori error estimates derived in Section 3.4 for both the hp-dgfem and the hp-sdfem.

23 3 (a) (b) Figure 5: Example. (a) Uniform 5 5 square mesh; (b) Quadrilateral mesh based on a 0% random perturbation of mesh (a). 4. Example In this example we let = ( ; ), a = (8=0; 6=0), b =, g = and f is chosen so that the analytical solution to (.) is given by u(x; y) = sin(( x)( y) =8); (4.) cf. []. We rst investigate the asymptotic behaviour of the hp-dgfem on a sequence of successively ner square and quadrilateral meshes for dierent p. In each case, the quadrilateral mesh is constructed from a uniform N N square mesh by randomly perturbing each of the interior nodes by up to 0% of the local mesh size: Figure 5 shows an example of a 5 5 square mesh together with the corresponding quadrilateral mesh. In Figure 6 we rst present a comparison of the DG-norm of the error with the mesh function h for p ranging between and 5. Here, we clearly see that kju u DG kj DG converges like O(h p= ) as h tends to zero for each (xed) p. Secondly, we investigate the convergence of the DGFEM with p{enrichment for xed h. Since the true solution (4.) is a real analytic function, we expect to observe exponential rates of convergence, cf. Remark 3.. Indeed, Figure 7 clearly illustrates this behaviour: on the linear{log scale, the convergence plots for each p become straight lines as the degree of the approximating polynomial is increased. Furthermore, we observe from Figures 6 & 7 that the h{ and p{convergence, respectively, of the DGFEM is robust with respect to mesh distortion. Finally, we verify the a priori error bound (3.) for the hp-sdfem. In Figures 8 & 9 we show the convergence of the scheme with respect to both h{ and p{renement, respectively. As with the DGFEM, we again observe optimal rates of convergence as h tends to zero for xed p (Figure 8) and exponential rates of convergence for xed h as p is increased (Figure

24 4 kju u DG kj DG p = p = p = 3 p = 4 p = h Square elements Quad. elements Figure 6: Example. Convergence of the DGFEM with h{renement kju u DG kj DG Square elements Quad. elements 5 5 mesh 9 9 mesh 7 7 mesh mesh p Figure 7: Example. Convergence of the DGFEM with p{renement.

25 5 kju u SD kj SD p = p = p = 3 p = 4 p = h Square elements Quad. elements Figure 8: Example. Convergence of the SDFEM with h{renement kju u SD kj SD Square elements Quad. elements 5 5 mesh 9 9 mesh 7 7 mesh mesh p Figure 9: Example. Convergence of the SDFEM with p{renement.

26 6 Figure 0: Example. 9 9 quadrilateral mesh aligned with the discontinuity. 9) on both uniform square meshes and quasi-uniform quadrilateral meshes. We remark that in all the computations performed here, the DGFEM was marginally more accurate than the SDFEM for each h and p; though, of course, the number of degrees of freedom in the DGFEM is greater than in the SDFEM for a given h and p. 4. Example In this example we let = ( ; ), a = (; 9=0), b = and f is chosen so that the analytical solution to (.) is given by sin((x ) =4) sin((y 9x=0)=) for x ; 9x=0 < y ; u(x; y) = e 5(x (y 9x=0) ) for x ; y < 9x=0; thus, u is discontinuous along the line y = 9x=0. To demonstrate the advantage of using discontinuous elements, we now only consider N N quadrilateral meshes which are aligned with the discontinuity; choosing N to be odd ensures that the discontinuity lies on element interfaces, cf. Figure 0. In this case the DGFEM does not `see' the lack of regularity in the problem and behaves as if the analytical solution u were smooth; i.e. optimal algebraic rates of convergence are observed with h{ renement and exponential rates of convergence are observed with p{renement. These results are summarized in Figure, where we show kju u DG kj DG in terms of the number of degrees of freedom. Thus, in practice, if an adaptive renement strategy is implemented which is capable of aligning the mesh with localised structures in the solution such as shocks, cf. [6] for example, then optimal, and indeed exponential, rates of convergence will be attained with the DGFEM. In contrast, from Figure we observe that the convergence rate of the SDFEM is limited by the regularity of u; we remark that by aligning the mesh with the discontinuity improves the accuracy of the SDFEM, though the rate of convergence of the scheme with h{ and p{renement is not enhanced.

27 p = kju u DG kj DG mesh 9 9 mesh 7 7 mesh p = p = 3 p = h{renement p{renement mesh p = Degrees of freedom Figure : Example. Convergence of the DGFEM with hp{renement on quadrilateral meshes aligned with the discontinuity. 0 0 kju u SD kj SD mesh p = 9 9 mesh p = 7 7 mesh 3 p = 3 p = mesh 3 p = 5 h{renement p{renement Degrees of freedom Figure : Example. Convergence of the SDFEM with hp{renement on quadrilateral meshes aligned with the discontinuity.

28 8 Finally, we note that if the mesh is not aligned with the discontinuity, then the DGFEM convergences at the same (slow) rate as the SDFEM; though, in all the numerical computations performed here, the DGFEM was marginally more accurate than the SDFEM for each h and p, cf. Example. References [].S. Bey and J.T. Oden, hp-version discontinuous Galerkin methods for hyperbolic conservation laws. Comput. Methods Appl. Mech. Engrg. 33 (996), pp. 59{86. [] D. Braess and C. Schwab, Approximation on triangles with respect to weighted Sobolev norms. (In preparation). [3] B. Cockburn and C. Schwab, hp-error analysis for the local discontinuous Galerkin method. (In preparation). [4] B. Cockburn, S. Hou, and C.-W. Shu, TVB Runge-utta local projection discontinuous Galerkin nite elements for hyperbolic conservation laws. Math. Comp. 54 (990), pp [5] L. Demkowicz,. Gerdes, C. Schwab, A. Bajer, and T. Walsh. HP90: A general and exible Fortran 90 hp-fe code. HP 90. Report 97-7, SAM, ETH-urich. Computing and Visualization in Science. (To appear). Available from the URL ftp://ftp.sam.math.ethz.ch/pub/sam-reports/reports/reports97/97-7.ps. [6] P. Houston, J. Mackenzie, E. Suli, and G. Warnecke, A posteriori error analysis for numerical approximations of Friedrichs systems. Numerische Mathematik. (To appear). Available from the URL friedrichs.ps.gz [7] C. Johnson, U. Navert, and J. Pitkaranta, Finite Element Methods for linear hyperbolic problems. Comp. Meth. Appl. Mech. Engrg. 45 (984) pp. 85{3. [8] C. Johnson and J. Pitkaranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic conservation law. Math. Comp. 46 (986) pp. {3. [9] P. Lesaint and P.A. Raviart, On a nite element method for solving the neutron transport equation. In: Mathematical aspects of Finite Elements in Partial Dierential Equations, C.A. deboor (Ed.), Academic Press New York (974), pp. 89{3. [0] C. Schwab, p- and hp-finite Element Methods. Theory and Applications to Solid and Fluid Mechanics. Oxford University Press, 998.