Adaptive Dis ontinuous Galerkin Finite Element Methods forCompressible Fluid FlowsPaul HoustonUniversity of Lei esterRalf HartmannUniversity of HeidelbergEndre S�uliUniversity of Oxford

1 Introdu tionMulti-dimensional ompressible uid ows are modelled by nonlinear onservation laws whosesolutions exhibit a wide range of lo alised stru tures, su h as sho k waves, onta t dis ontinu-ities and rarefa tion waves. The a urate numeri al resolution of these features ne essitatesthe use of lo ally re�ned, adaptive omputational meshes. A majority of adaptive algorithmsused to model ompressible uid ows will simply re�ne or adjust the omputational grida ording to an ad ho riterion, su h as a large gradient in a physi al quantity, for exam-ple. Although this intuitive approa h has had some su ess, it does not provide guaranteederror ontrol. Moreover, ad ho re�nement strategies may not provide the most e onomi almesh design for the ontrol of a given error quantity of interest, f. [2, 3℄, for example. Theaim of this paper is to dis uss the a posteriori error analysis and adaptive mesh design fordis ontinuous Galerkin �nite element approximations to systems of onservation laws. InSe tion 2, we introdu e the model problem and formulate its dis ontinuous Galerkin �niteelement approximation. Se tion 3 is devoted to the derivation of weighted a posteriori errorbounds for linear fun tionals of the solution. Finally, in Se tion 4 we present some numeri alexamples to demonstrate the performan e of the resulting adaptive �nite element algorithm.2 Model problem and dis retisationGiven an open bounded polyhedral domain in R n , n � 1, with boundary �, we onsiderthe following problem: �nd u : ! Rm , m � 1, su h thatdivF(u) = 0 in ; (2.1)where, F : Rm ! Rm�n is ontinuously di�erentiable. We assume that the system of on-servation laws (2.1) may be supplemented by appropriate initial/boundary onditions. Forexample, assuming that B(u; �) := Pni=1 �iruFi(u) has m real eigenvalues and a ompleteset of linearly independent eigenve tors for all � = (�1; : : : ; �n) 2 R n ; then at in ow/out owboundaries, we require that B�(u;n)(u� g) = 0; where n denotes the unit outward normalve tor to �, B�(u;n) is the negative part of B(u;n) and g is a (given) real{valued fun tion.To formulate the dis ontinuous Galerkin �nite element method (DGFEM, for short) for(2.1), we �rst introdu e some notation. Let Th = f�g be an admissible subdivision of intoopen element domains �; here h is a pie ewise onstant mesh fun tion with h(x) = diam(�)1

2 Houston et al.when x is in element �. For p 2 N 0 , we de�ne the following �nite element spa eSh;p = fv 2 [L2()℄m : vj� 2 [Pp(�)℄m 8� 2 Thg ; (2.2)where Pp(�) denotes the set of polynomials of degree at most p over �. Given that v 2�H1(�)�m for ea h � 2 Th, we denote by v+ (resp., v�) the interior (resp., exterior) tra e ofv on ��. The DGFEM for (2.1) is de�ned as follows: �nd uh 2 Sh;p su h thatX�2Th

��Z�F(uh) � rvh dx+ Z��H(u+h ;u�h ;n�)v+h ds+ Z� "ruh � rvh dx� = 0 (2.3)for all vh 2 Sh;p, f. [2, 5℄, for example. Here, H(�; �; �) denotes a numeri al ux fun tion,assumed to be Lips hitz ontinuous, onsistent and onservative. We emphasize that the hoi e of the numeri al ux fun tion is ompletely independent of the �nite element spa eemployed; in Se tion 4, we employ the (lo al) Lax{Friedri hs ux. Further, " denotes the oeÆ ient of arti� ial vis osity de�ned by " = C" h2�� jdivF(uh)j, where C" is a positive onstant and 0 < � < 1=2; see [5℄. For elements � 2 Th whose boundary interse ts �, werepla e u�h by appropriate boundary/initial onditions on �� \ �.3 A posteriori error analysisIn this se tion, we shall be on erned with ontrolling the error in the numeri al solutionmeasured in terms of a given linear fun tional J(�). Here, J(�) is the physi al quantity ofinterest whi h may, for example, represent the mean ow a ross a urve, a point value of thesolution, or the lift or drag oeÆ ients of a body immersed into an invis id uid, f. [3℄.We begin by �rst introdu ing some notation. Letting N (uh;vh) denote the left{hand sideof (2.3), we write M(u;uh; �; �) to denote the mean{value linearisation of N (�; �) given by

M(u;uh;u� uh;v) = N (u;v)�N (uh;v) = Z 10 N 0u[�u+ (1� �)uh℄(u� uh;v) d� (3.1)for all v in V , where V is a suitable fun tion spa e su h that Sh;p � V . Here, N 0u[w℄(�;v)denotes the Fr�e het derivative of u 7! N (u;v), for v 2 V �xed, at some w in V . Weremark that the linearisation de�ned in (3.1) is only a formal al ulation, in the sense thatN 0u[w℄(�; �) may not in general exist. Instead, a suitable approximation to N 0u[w℄(�; �) mustbe determined, for example, by omputing appropriate �nite di�eren e quotients of N (�; �), f. [2℄. For the pro eeding analysis, we assume that the linearisation (3.1) is well{de�ned.Under this hypothesis, we introdu e the following dual problem: �nd z 2 V su h thatM(u;uh;w; z) = J(w) 8w 2 V: (3.2)We assume that (3.2) possesses a unique solution. Clearly, the validity of this assumptiondepends on both the de�nition of M(u;uh; �; �) and the hoi e of the linear fun tional under onsideration, f. [2℄. For the pro eeding error analysis, we must therefore assume that thedual problem (3.2) is well{posed. Under this assumption, we have the following result.Theorem 1 Let u and uh denote the solutions of (2.1) and (2.3), respe tively, and supposethat the dual problem (3.2) is well{posed. Then,J(u)� J(uh) = E � X�2Th ��; (3.3)

Adaptive Dis ontinuous Galerkin Finite Element Methods for Compressible Fluid Flows 3where �� = Z�Rh (z� zh) dx+ Z�� rh (z� zh)+ ds� Z� "ruh � r(z� zh) dxfor all zh in Sh;p. Here, Rhj� = �divF(uh) and rhj� = F(uh) � n� � H(u+h ;u�h ;n�) denotethe internal and boundary �nite element residuals, respe tively, de�ned on ea h � 2 Th.Proof Choosing w = u � uh in (3.2), re alling the linearity of J(�), and exploiting the Galerkinorthogonality property N (u;vh)�N (uh;vh) = 0 for all vh in Sh;p, we getJ(u)� J(uh) = J(u� uh) =M(u;uh;u� uh; z) =M(u;uh;u� uh; z� zh) = �N (uh; z� zh)for all zh in Sh;p. Equation (3.3) now follows by employing the divergen e theorem. �Based on the general error representation formula derived in Theorem 1, a posteriori errorestimates bounding the error in the omputed fun tional J(�) may be dedu ed. Before wepro eed, let us �rst re all the lassi� ation into Type I and Type II a posteriori error boundsintrodu ed in [4℄. Type I a posteriori error bounds, also referred to as weighted a posteriorierror bounds, f. [3℄, involve the multipli ation of the omputable terms Rh, rh and "ruh, bythe di�eren e between the dual solution z and its proje tion (or interpolant) zh. On the otherhand, Type II a posteriori bounds are in the spirit of the error analysis of Johnson et al. [1℄,and do not depend expli itly on the dual solution. These latter error estimates are derivedfrom Type I a posteriori bounds by employing standard results from approximation theory toestimate the proje tion (or interpolation) error, together with well{posedness results for thedual problem. The resulting Type II error bounds only involve ertain norms of the residualsand arti� ial vis osity term, the mesh fun tion h, interpolation onstants and the stabilityfa tor of the dual problem.However, as we shall see in Se tion 4, the elimination of the `weighting terms' involvingthe di�eren e between z and zh may adversely a�e t the performan e of our adaptive �niteelement method. Indeed, mesh re�nement strategies based on Type II a posteriori errorbounds whi h do not require the omputation of the dual solution may, under mesh re�ne-ment, lead to suboptimal onvergen e of the error J(u)� J(uh) in the omputed fun tional,resulting in une onomi al mesh design and an ineÆ ient adaptive algorithm, f. [3℄. For thisreason, we only onsider the derivation of the following Type I a posteriori error bound.Corollary 1A Under the assumptions of Theorem 1, we havejJ(u)� J(uh)j � Ejj � X�2Th j��j : (3.4)Proof The error bound (3.4) follows from (3.3) by appli ation of the triangle inequality. �We end this se tion by noting that for nonlinear hyperboli onservation laws, both theerror representation formula (3.3) and the Type I a posteriori error bound (3.4) depend onthe unknown analyti al solution to the primal and dual problems. Thus, in order to renderthese quantities omputable, both u and z must be repla ed by suitable approximations.To this end, the linearisation leading to M(u;uh; �; �) is performed about uh and the dualsolution z is repla ed by a dis ontinuous Galerkin approximation z omputed on the samemesh Th used for uh, but with a higher degree polynomial, i.e., z 2 Sh;p, p > p.4 Numeri al experimentsIn this se tion we present some numeri al examples to highlight the advantages of designingan adaptive �nite element algorithm based on the weighted error indi ators j��j in omparison

4 Houston et al.Nodes Elements DOF J(u� uh) P� �� �1 P� j��j �281 128 384 0.2441E-01 0.4158E-01 1.70 0.9703E-01 3.97135 230 690 -0.2797 0.4037E-01 -0.14 3.750 13.41254 464 1392 -0.2507 -0.3753E-01 0.15 0.2777 1.11566 1079 3237 -0.1173 -0.4558E-01 0.39 0.8735E-01 0.741100 2142 6426 0.3788E-03 0.3023E-03 0.80 0.2829E-02 7.472005 3942 11826 0.1801E-04 0.1877E-04 1.04 0.4967E-03 27.583624 7180 21540 0.2559E-05 0.2476E-05 0.97 0.2387E-03 93.276938 13808 41424 0.3654E-06 0.3650E-06 1.00 0.1045E-03 286.0113560 27052 81156 0.8224E-07 0.8445E-07 1.03 0.4244E-04 516.1127209 54343 163029 0.4271E-08 0.4551E-08 1.07 0.1718E-04 4023.63

Table 1: Example 1. Adaptive algorithm for Burgers' equation.with traditional re�nement strategies whi h do not require the solution of the dual problem(3.2). To this end, we onsider the ad ho error indi ator �adho � = khRhkL2(�)+kh1=2rhkL2(��),whi h stems from a Type II a posteriori error analysis, f. [4℄. Here, we employ the �xedfra tion mesh re�nement algorithm with re�nement and dere�nement fra tions set to 20%and 10%, respe tively. Finally, we note that all omputations are performed with p = 1 andp = 2.4.1 Burgers' equationHere, we onsider the one{dimensional unsteady invis id Burgers' equation for the s alarvariable u � u; i.e., writing x2 to denote time, we have ux2 + �(1=2)u2�x1 = 0; on the(spa e{time) domain = (0; 1)2, subje t to the initial ondition

u(x1; 0) =8>><>>:3=2; for x1 � 0;0; for 0 < x1 � 1=8;(3=8)( os(4�(x1 � 3=8)) + 1); for 1=8 < x1 � 5=8;0; for x1 > 5=8;and boundary ondition u(0; x2) = 3=2, for x2 2 [0; 1℄. The analyti al solution to this problem onsists of two sho ks in the spa e{time domain separated by a smooth region ontaining ararefa tion wave. The two sho ks eventually merge to form a single line of dis ontinuity inthe spa e{time plane. In this example we sele t the fun tional of interest J(�) to be the valueof the solution just before the sho ks intera t with one another. More pre isely, we hooseJ(u) = u(0:8; 0:9); thereby, the true value of the fun tional is J(u) = 0:564318816016671.In Table 1, we demonstrate the performan e of the adaptive algorithm with C" = 1=4 and� = 1=10. Here, we show the number of nodes, elements and degrees of freedom (DOF) inSh;1, the true error in the fun tional J(u � uh), the omputed error representation formula(3.3), the a posteriori error bound (3.4) and their respe tive e�e tivity indi es �1 and �2.We see that initially on very oarse meshes the quality of the omputed error representationformula E is very poor, in the sense that �1 = E=J(u� uh) is not lose to one; however, asthe mesh is re�ned the e�e tivity index �1 approa hes unity. On the other hand, we observethat the Type I a posteriori error bound is not sharp, in the sense that the se ond e�e tivityindex �2 overestimates the true error in the omputed fun tional by three orders of magnitudeon the �nest mesh; this loss of sharpness is attributed to the loss of inter{element an ellationof the lo al error indi ators ��, when the triangle inequality is employed. Thereby, it is lear

Adaptive Dis ontinuous Galerkin Finite Element Methods for Compressible Fluid Flows 5

(a) (b)Figure 1: Example 1. (a) Mesh onstru ted using dual error indi ator with 27209 nodes and54343 elements (jJ(u) � J(uh)j = 4:271 � 10�9); (b) Mesh onstru ted using ad ho errorindi ator with 17288 nodes and 34432 elements (jJ(u)� J(uh)j = 1:358� 10�5).

102

103

104

105

106

10−10

10−8

10−6

10−4

10−2

100

Dual Indicator Ad hoc Indicator

PSfrag repla ements jJ(u)�J(u h)j

Degrees of FreedomFigure 2: Example 1. Convergen e of J(u)�J(uh) using the dual and ad ho error indi ators.that any further bounding performed en route to deriving a Type II a posteriori estimate willfurther adversely a�e t the quality of the omputed error bound.In Figure 1 we show the meshes generated using both the weighted error indi ator j��jand the ad ho error indi ator �adho � . From Figure 1(a), we see that there is virtually no re-�nement in the regions of the omputational domain where the sho ks are lo ated when j��jis employed. Indeed, the most of the mesh re�nement is on entrated in the neighbourhoodupstream of the point of interest. In ontrast, the mesh produ ed using the ad ho error indi- ator is largely on entrated the vi inity of the two sho k waves, with some re�nement of thesmooth region between them. In Figure 2, we ompare the true error in the omputed fun -tional J(�) using the two mesh re�nement strategies. Here, we learly observe the superiorityof the weighted a posteriori error indi ators; on the �nal mesh the true error in the linearfun tional is over three orders of magnitude smaller than J(u�uh) omputed on the sequen eof meshes produ ed using �adho � . This learly indi ates that a good numeri al resolution ofthe sho ks is irrelevant for the a urate approximation of the fun tional of interest.

6 Houston et al.Nodes Elements DOF J(u� uh) P� �� �1 P� j��j �291 144 1728 -0.6734E-03 -0.6822E-03 1.01 0.6595E-02 9.79157 274 3288 0.1488E-03 0.1473E-03 0.99 0.2827E-02 19.00323 596 7152 0.3327E-04 0.3216E-04 0.97 0.1114E-02 33.47564 1060 12720 -0.1006E-04 -0.9445E-05 0.94 0.7351E-03 73.041010 1931 23172 -0.1250E-04 -0.1241E-04 0.99 0.3145E-03 25.161905 3702 44424 -0.8657E-05 -0.8545E-05 0.99 0.2257E-03 26.073706 7263 87156 -0.2951E-05 -0.2933E-05 0.99 0.8870E-04 30.057215 14228 170736 -0.1087E-05 -0.1079E-05 0.99 0.3792E-04 34.88

Table 2: Example 2. Adaptive algorithm for Ringleb's ow.

(a) (b)Figure 3: Example 2. (a) Mesh onstru ted using dual error indi ator with 7215 nodes and14228 elements (jJ(u) � J(uh)j = 1:087 � 10�6); (b) Mesh onstru ted using ad ho errorindi ator with 4627 nodes and 8965 elements (jJ(u)� J(uh)j = 2:332� 10�5).4.2 Ringleb's owIn this �nal example we onsider the steady two{dimensional ompressible Euler equations;here, u represents the ve tor of onserved quantities (�; �u; �v; �E), where �, (u; v) and Erepresent the density, Cartesian velo ity and total energy per unit mass, respe tively. Here, we onsider Ringleb's ow for whi h an analyti al solution may be obtained using the hodographmethod. This problem represents a transoni ow whi h turns around an obsta le; the owis mostly subsoni , with a small supersoni region around the nose of the obsta le.Given that the solution to Ringleb's ow is smooth, no arti� ial vis osity is required;thereby, we set C" = 0. Further, we take the fun tional of interest to be the value of thetotal energy at the point (�0:4; 2), i.e., J(u) = (�E)(�0:4; 2); onsequently the true value ofthe fun tional is given by J(u) = 1:574181283916596. In Table 2 we show the performan eof our adaptive algorithm; here we see that the quality of the omputed error representationformula is extremely good, with �1 � 1 even on very oarse meshes. Furthermore, the Type Ia posteriori error bound (3.4) is sharper for this smooth problem than for Example 1; indeed,here Ejj overestimates jJ(u� uh)j by just over an order of magnitude.The meshes produ ed using both our weighted error indi ator j��j and the ad ho error

Adaptive Dis ontinuous Galerkin Finite Element Methods for Compressible Fluid Flows 7

103

104

105

106

10−6

10−5

10−4

10−3

Dual Indicator Ad hoc Indicator

PSfrag repla ements jJ(u)�J(u h)j

Degrees of FreedomFigure 4: Example 2. Convergen e of J(u)�J(uh) using the dual and ad ho error indi ators.indi ator �adho � are shown in Figure 3. Here, we see that, as in the previous example, themesh onstru ted using j��j is mostly on entrated in the neighbourhood upstream of thepoint of interest. However, due to the ellipti nature of the ow in the subsoni region, a ir ular region ontaining the point of interest is also re�ned, together with a strip of ellson the wall of the obsta le en losing the supersoni region of the ow. In ontrast, the mesh onstru ted using �adho � is largely on entrated in the region near the obsta le, with furtheralmost uniform re�nement of the rest of the omputational domain; we note that subsequentmesh re�nement leads to an even greater on entration of elements near the obsta le. Finally,in Figure 4 we ompare the true error in the omputed fun tional J(�) using the two meshre�nement strategies. Again, as in Example 1, we learly see that our weighted a posteriorierror indi ators produ e more e onomi al meshes than when the traditional ad ho errorindi ator is employed, in the sense that J(u � uh) is smaller for a given number of degreesof freedom. Indeed, on the �nal mesh the true error in the omputed fun tional is over anorder of magnitude smaller when the former error indi ator is used.A knowledgmentsRalf Hartmann a knowledges the �nan ial support of the DFG Priority Resear h Programand the SFB 359 at the IWR, University of Heidelberg. Paul Houston a knowledges the�nan ial support of the EPSRC (Grant GR/N24230).Referen es[1℄ P. Hansbo and C. Johnson, Streamline di�usion �nite element methods for uid ow. vonKarman Institute Le tures, 1995.[2℄ R. Hartmann and P. Houston, Adaptive Dis ontinuous Galerkin Finite Element Methods forNonlinear Hyperboli Conservation Laws. In preparation.[3℄ P. Houston, R. Ranna her and E. S�uli, A posteriori error analysis for stabilised �nite ele-ment approximations of transport problems. Comput. Meth. Appl. Me h. Engrg. 190(11-12):1483{1508, 2000.[4℄ P. Houston and E. S�uli, hp{Adaptive dis ontinuous Galerkin �nite element methods for �rst{order hyperboli problems. Lei ester University Te hni al Report 2000/31, 2000. (Submitted).[5℄ J. Jaffre, C. Johnson and A. Szepessy, Convergen e of the dis ontinuous Galerkin �niteelement method for hyperboli onservation laws. Math. Models Methods Appl. S i. 5(3):367{386,1995.