A posteriori error estimates have been an integral part of ever adaptive method, thus for the DGM to be used in an adaptive setting one needs to devel

Transcription

1 A Posteriori Discontinuous Finite Element Error Estimation for Two-Dimensional Hperbolic Problems Slimane Adjerid and Thomas C. Masse Department of Mathematics Virginia Poltechnic Institute and State Universit Blacksburg, VA 246 June 9, 22 Abstract We analze the discontinuous finite element errors associated with p degree solutions for two-dimensional first-order hperbolic problems. We show that the error on each element can be split into a dominant and less dominant component and that the leading part is O(h p+ ) and is spanned b two (p +) degree Radau polnomials in the and directions, respectivel. We show that the p degree discontinuous finite element solution is superconvergent at Radau points obtained as a tensor product of the roots of (p +) degree Radau polnomial. For a linear model problem, the p degree discontinuous Galerkin solution flu ehibits a strong O(h 2p+2 ) local superconvergence on average at the element outflow boundar. We further establish an O(h 2p+ ) global superconvergence for the solution flu at the outflow boundar of the domain. These results are used to construct simple, efficient and asmptoticall correct a posteriori finite element error estimates for multi-dimensional first-order hperbolic problems in regions where solutions are smooth. Introduction The discontinuous Galerkin method (DGM) was first used for the neutron equation b Reed et al. [2]. Since then, DGM methods have been used to solve hperbolic [4, 5, 6, 7, 2,, 3, 6], parabolic [4, 5], and elliptic [3, 2, 23] partial differential equations. For a more complete list of citations on the DGM and its applications consult []. The discontinuous Galerkin method has been used successfull to solve sstems of hperbolic sstems of conservation laws. The DGM method has several advantages over continuous methods, since it uses completel discontinuous polnomial bases it can sharpl capture discontinuities which are common for hperbolic problems and makes order variations and mesh refinement much easier in the presence of hanging nodes. The DGM has a simple communication pattern that makes it useful for parallel computations.

2 A posteriori error estimates have been an integral part of ever adaptive method, thus for the DGM to be used in an adaptive setting one needs to develop a posteriori error estimates that will be used to guide the adaptive process b indicating regions where more or less resolution is needed, to assess the qualit of the solution and to stop the adaptive process. A variet of a posteriori error estimates for DG methods have been developed b Be and Oden [4], Suli [22] and Riviere and Wheeler [2] for linear problems. For nonlinear problems consult [8, 9, 9]. Adjerid et al. [] proved that smooth DGM solutions of one-dimensional hperbolic problems using p-degree polnomial approimations have an O(h p+2 ) superconvergence rate at the roots of Radau polnomial of degree p +. The used this result to construct asmptoticall correct a posteriori error estimates. The further established a strong O(h 2p+ ) superconvergence at the downwind end of ever element. Krivodonova et al. [8] proved a superconvergence result on average on the outflow edge of ever element of unstructured triangular meshes and constructed aposteriori error estimates that converge to the true error under mesh refinement. In this paper we etend the one-dimensional results of Adjerid et al. [] to multidimensional hperbolic problems on rectangular meshes. We show how to select the finite element space such that the leading term in the true local error is spanned b two (p + ) degree Radau polnomials in the and directions, respectivel. The p- degree discontinuous finite element solution ehibits an O(h p+2 ) superconvergence at the Radau points obtained as a tensor product of the roots of Radau polnomial of degree p +. For a linear model problem we show that, locall, the solution flu is O(h 2p+2 ) superconvergent on average on the outflow element boundar. We further show that the solution flu converges at an O(h 2p+ ) rate on average at the outflow boundar of the domain. We use these superconvergence results to construct asmptoticall correct a posteriori error estimates. The paper is organized as follows, in 2 we present the error analsis and new superconvergence results. In 3 we construct several aposteriori error estimates for linear and nonlinear problems. We present computational results for several linear and nonlinear problems in 4. The computational results suggest that the theor of 2 etends to more general situations such as nonlinear conservation laws and solutions with discontinuities, ecept for the flu superconvergence which is under further investigation. We conclude and discuss our results in 5. 2 Error analsis We begin our error analsis b considering the linear first-order model problem ff:ru = f(; ); (; ) 2 Ω=[; ] [; ]; (2.a) subject to the boundar conditions at the inflow boundaries u(; ) = g () and u(;)=g (): (2.b) 2

3 in out denote the boundar of Ω and ν be the outward unit normal The inflow boundar in = f(; ) j ff ν < g (2.2a) and the outflow boundar out = f(; ) j ff ν > g: (2.2b) We assume that ff = [ff ;ff 2 ] T is constant with ff i > ; i = ; 2: The functions f(; ), g () and g () are selected such that the eact solution u(; ) 2 C (Ω). In order to obtain a weak discontinuous Galerkin formulation, we partition the domain Ω into N = n n square elements and start the integration with elements whose inflow boundar is on the domain inflow boundaries. In order to perform an error analsis we consider the first element = [;h] 2 where h = =n and the space V p of polnomial functions such that P p+ ρv p [f p+ ; p+ g; p ; (2.3a) where P k is the space of polnomials of degree k P k = fq j q = kx mx m= i= c m i i m i g: (2.3b) These spaces are suboptimal but the lead to a ver simple a posteriori error estimator. For efficienc reasons we consider the smallest spaces that satisf (2.3) V p = f V j V = px k= kx i= c k i i k i + px i= c p+ i i p+ i g: (2.4) We note that tensor product elements satisf (2.3). The discontinuous Galerkin method [7] consists of determining U(; ) 2V p on such that ff νv (U U)dff + ff ruvdd = f(; )V dd; 8 V 2V p ; (2.5) where = [ is the boundar of, with and, respectivel, denoting the inflow and outflow boundaries of as shown in Figure. For the boundar data U on we use either the eact boundar condition or its interpolant ( U ßg ; if (; ) 2 (; ) = ; (2.6) ßg ; if (; ) 2 4 3

4 Γ 3 Γ 4 Γ 2 [α, α ] T 2 Γ Figure : An element with inflow (solid) and outflow (dashed) boundaries. where ßw is the p-degree polnomial that interpolates w at the roots of (p + )-degree right Radau polnomial R p+ (ο) =L p+ (ο) L p (ο);» ο» ; (2.7) with L p being Legendre polnomial of degree p. Once we determine the solution on the first element we proceed to the elements whose inflow boundaries are either on the inflow boundar of Ω or an outflow boundar of and continue this process until the solution is determined in the whole domain. On an element whose inflow boundar is not on the boundar of Ω, U is defined as U (; ) =lim s! U((; )+sν); (; ) 2 : (2.8) We will show that the discontinuous Galerkin local finite element error can be split into an O(h p+ ) component which is a linear combination of two right Radau polnomials of degree p+ in the and directions, respectivel, and a higher-order component. Prior to this, we need to establish the DGM orthogonalit condition and prove two preliminar lemmas. We begin b multipling (2.) b V 2V p, integrating over the element and using Green's formula to obtain ff νv udff ff rvudd = f(; )V dd: (2.9) Appling Green's formula to (2.5) leads to ff νuvdff + ff νu Vdff ff rvudd = 4

5 f(; )V dd: (2.) Subtracting (2.) from (2.9) leads to the DGM orthogonalit condition where ff ν ffl Vdff+ ff ν fflv dff h h ff rvffldd =; 8V 2V p ; (2.) ffl = u U (2.2) is the local finite element discretization error. Using the mapping of [;h] 2 onto the canonical element [ ; ] 2 defined b = h( + ο)=2 and = h( + )=2 and ^u(ο; ) = u((ο);( )) we obtain the DGM orthogonalit condition (2.) on the canonical element ff ν^ffl ^ in ^Vd^ff + ff ν^ffl ^Vd^ff ff r^v ^ffldο d =; 8 ^V 2 ^V p : (2.3) ^ In the remainder of this paper we will omit the^unless we feel it is needed for clarit. Lemma 2.. If Q k 2V k satisfies then ff νq k Vdff Proof. Using Green's formula we write (2.4) as ff νq k Vdff+ ff rvq k dο d =; 8V 2V p ; k» p; (2.4) Q k =; k» p: (2.5) Adding (2.4) to (2.6) with V = Q k we obtain ff νq 2 kdff + ff νq 2 kdff = ff rq k Vdο d =; 8V 2V p : (2.6) jff νjq 2 kdff =; k» p: (2.7) This leads to Q k = on. Combining this with (2.6) for V = ffrq k ields ffrq k = on which completes the proof. In the following lemma we will write the interpolation error as a power series in h. Lemma 2.2. Let w 2 C (;h) and ßw be a p-degree polnomial that interpolates w at Radau points on [,h]. Then the interpolation error w((ο)) ßw((ο)) = X k=p+ Q k (ο)hk ; (2.8) 5

6 where and Q p+(ο) = w(p+) () 2 p+ (p + )! (ο ο )(ο ο ) :::(ο ο p )=c p+ R p+ (ο); (2.9) with r k (ο) being a polnomial of degree k. Q k (ο) =R p+(ο)r k p (ο); k>p+; (2.2) Proof. B the standard interpolation error there eists s() such that w() ßw() = w(p+) (s(; h)) ( )( ) :::( p ); 2 [;h]; (2.2) (p + )! where i = h( + ο i )=2 and ο i ;i = ; ; ::: ; p; are the roots of right Radau polnomial R p+ in [-,]. On [ ; ] the interpolation error can be written as w(ο) ßw(ο) = hp+ w (p+) (s((ο);h)) (ο ο 2 p+ )(ο ο ) :::(ο ο p ); ο 2 [ ; ]: (2.22) (p + )! The Maclaurin series of w (p+) (s((ο);h)) with respect to h ields where w (p+) (s((ο);h)) = w (p+) () + X k= h k q k (ο) (2.23a) q k (ο) = d k w (p+) (s((ο);h)) k! dh k j h= ; k>; (2.23b) is a polnomial of degree k. Combining (2.22) and (2.23) completes the proof. Now we are read to state the main result of this section. Theorem 2.. Let u and U be the solution of (2.) and (2.5), respectivel. Then the local finite element error can be written as where ffl(ο; ) = X k=p+ h k Q k (ο; ); (2.24) Q p+ = fi R p+ (ο)+fi 2 R p+ ( ): (2.25) Furthermore, at the outflow boundar of the phsical element ff ν ffldff = O(h 2p+2 ): (2.26) 6

7 Proof. Writing the Maclaurin series of the local error ffl with respect to h ields ffl = X k= Q k (ο; )h k : (2.27) Substituting the series (2.27) in the DGM orthogonalit condition (2.3), using (2.8) and collecting terms having the same powers of h we write X k=p+ h k px k= h k ff:νq k Vdff ff νq k Vdff+ ff νq k Vdff ff rvq k dο d + ff rvq k dο d =; 8V 2V p : (2.28) Thus, for» k» p we have ff νq k Vdff ff rvq k dο d =; 8V 2V p : (2.29) B lemma 2., (2.29) leads to Q k =; k = ; ; ::: ; p; and establishes (2.24). On the other-hand, using Lemma 2.2, we write Q p+(ο; ) = 8 >< >: 2 p+ (p+)! 2 p+ p+ p+ (; )(ο ο )(ο ο ) :::(ο ο p ); on A direct computation reveals that Q p+ can be split as Q p+ = where ~Q p (ο; ) 2V p and : p+ u(; )( p+ )( ο ) :::( ο p ); on 4 d p+ (u U) (ο; )j (p + )! dh p+ h= = Q» p+ + Q ~ p ; (2.3a)»Q p+ p+ u 2 p+ (p + (; )(ο ο )(ο ο p+ ) :::(ο ο p p+ u + 2 p+ (p + (; )( ο )( ο p+ ) :::( ο p ): (2.3b) Substituting (2.3) in the O(h p+ ) term of the series (2.28) leads to ff νq p+vdff+ ff ν»q p+ Vdff 7 ff rv»q p+ dο d

8 + ff ν Q ~ p Vdff Using (2.3) and (2.3b) we can show that ff νq p+vdff+ ff ν» Q p+ Vdff ff rv Q ~ p dο d =; 8V 2V p : (2.32) Combining (2.32) and (2.33) with Lemma 2. leads to ~Q p (2.25). Using (2.3) with Lemma 2.2 we can show that ff rv Q» p+ dο d = ; 8V 2V p : (2.33) =. Using (2.3) we establish ff νq k Vdff = ; 8V 2V 2p k; k = p +; ; 2p: (2.34) Using (2.34), the O(h k ); p +» k» 2p, term of (2.28) ields ff νq k Vdff Testing against V = we obtain which establishes (2.26). ff rvq k dο d =; 8V 2V 2p k : (2.35) ff νq k dff =; k = p +; ; 2p; (2.36) In the previous theorem we have etended the results of Adjerid et al. [] to multidimensional problems on square meshes. In particular, we have proved that the p degree DG solution is O(h p+2 ) superconvergent at Radau points (ο i ;ο j ); i; j =; :::; p and that the DG solution has an O(h 2p+2 ) superconvergence on average at the outflow boundar of. We note that for an arbitrar nonzero constantvector ff, (2.24), (2.25) and (2.26) hold where the significant part of the error is spanned b the p + -degree Radau polnomials R p+ (ο) =L p+ (ο) sign(ff )L p (ο); R p+ ( ) =L p+ ( ) sign(ff 2 )L p ( ): (2.37) 2. Global error analsis In the following theorem we will show a superconvergence result for the global discretization error if p. Theorem 2.2. Under the conditions of Theorem 2. the global finite element error e for p satisfies the superconvergence result Ω out ff νe(; ) dff = O(h 2p+ ): (2.38) 8

9 Proof. Adding and subtracting u(; ) to both terms on the left side of (2.5) we obtain ff νv (U u + u U)dff + ff r(u u + u)v dd = f(; )V dd; 8 V 2V p ; (2.39) where is an arbitrar element. This can be written as ff νv (e e )dff ff rev dd = 8 V 2V p : (2.4) Integrating b parts leads to ff νve dff + ff νv edff ff rvedd = Testing against V = ields ff νe dff + Summing over all elements we obtain Ω in ff νe dff + Combining ( 2.43) and Lemma 2.2 we establish (2.38). 2.2 Nonlinear problems We will describe similar results for problems of the form 8 V 2V p : (2.4) ff νedff =: (2.42) Ω out ff νedff =: (2.43) ff ru + ffi(u) =f(; ); (; ) 2 Ω; (2.44) with boundar conditions at the inflow boundar. The DGM weak formulation consists of determining U(; ) 2V p on such that ff νv (U U)dff + [ff ruv + ffi(u)v ]dd = f(; )V dd; 8 V 2V p : (2.45) In the following theorem we state superconvergence results and local error estimates for nonlinear problems. 9

10 Theorem 2.3. Let u and U be the solution of (2.44) and (2.45), respectivel. If u is smooth, then the local error estimates (2.24) and (2.25) hold. Proof. The DGM orthogonalit for (2.44) on an element can be written as ff νffl Vdff+ ff νfflv dff h h [ff rvffl+(ffi(u) ffi(u))v ]d d =; On [ ; ] [ ; ] the orthogonalit condition (2.46) becomes ff νffl Vdff+ ff νfflv dff 8V 2V p : (2.46) [ff rvffl+ h 2 (ffi(u) ffi(u))v ]dο d =; 8V 2V p : (2.47) Assuming ffi analtic and ffi bounded and using Talor series of ffi about u we have ffi(u) ffi(u) =a(u)ffl ffl2 2 ffi (μu); a(u) =ffi (u): (2.48a) The Maclaurin series of a(u) with respect to h ields a(u) =2 X k= h k μq k (ο; ); μq k (ο; ) = 2 ffi (k+) (u((ο);( ))) d k u k! dh k ((ο);( ))j h= 2P k : (2.48b) We note that in order to establish (2.48b) we used the chain rule with (ο) =h( + ο)=2 and ( ) =h( + )=2. Substituting (2.27) and (2.48) in (2.47) and collecting terms having same powers of h lead to ff:νq Vdff ff rvq dο d + px k= h k ff:νq k Vdff [ff rvq k + k V ]dο d out X h k ( ff νq in k Vdff+ ff νq k Vdff k=p+ [ff rvq k + k V ]dο d ) =; 8V 2V p ; (2.49a) +

11 where k = kx μq l Q k l : (2.49b) l= Appling Lemma 2.2, the O() term ields Q =. B induction we prove that Q k = ; k =; ;::: ;p. We note that the term in (2.48a) involving ffl 2 is higher order and does not contribute to our leading terms. Following the same lines of reasoning used to prove (2.25) we establish the same result for nonlinear problems. 3 A Posteriori error estimation The results of Theorem 2. and (2.2) suggest that the global finite element error on each element can be approimated as e(; ) = u U ß E(; ) = b R p+ ()+b 2 R p+ (): (3.) Substituting u in (2.5) b e + U leads to ff r(u + e)w dd = fw dd: (3.2) Approimating e b E ields the following discrete problem for the error ff r(u + E)R p+ ()dd = ff r(u + E)R p+ ()dd = fr p+ ()dd; fr p+ ()dd: (3.3a) (3.3b) We note that the strong superconvergence (2.26) of p-degree, p>, finite element solution flu at the outflow boundar of each element suggests that we should neglect the jump terms in the error problem without compromising the accurac of our error estimate. Since for p = this strong superconvergence at the outflow boundar is lost and (3.3) fails to have a unique solution, we solve the following problem for the error estimate ff ν(u + E U E)R ()dff + = fr ()dd; ff ν(u + E U E)R ()dff + ff r(u + E)R ()dd ff r(u + E)R ()dd (3.4a)

12 For nonlinear problems of the form = fr ()dd: (3.4b) we find E b solving the linearized problem [h (U); ] r(u + E)R p+ ()dd = u + h(u) = f(; ) (3.5) [h (U); ] r(u + E)R p+ ()dd = fr p+ ()dd; fr p+ ()dd: (3.6a) (3.6b) We also use the solution of the linearized problem (3.6) as an initial guess for Newton iteration when solving the nonlinear finite element problem for E [h (U + E); ] r(u + E)R p+ ()dd = [h (U + E); ] r(u + E)R p+ ()dd = fr p+ ()dd; fr p+ ()dd: (3.7a) (3.7b) An accepted efficienc measure of a posteriori error estimates is the effectivit inde. In this paper we use the local effectivit indices in the L 2 norm and the global effectivit inde i = jjejj L 2 (i) ; i =; 2; ::: ; N (3.8) jjejj L 2 (i) = jjejj L 2 (Ω) : (3.9) jjejj L 2 (Ω) Ideall, effectivit indices should approach unit under mesh refinement. 4 Numerical eamples Eample. We consider the linear hperbolic problem u +2u = f(; ); (; ) 2 [; ] 2 (4.a) subject to the boundar conditions u(; ) = g (); u(;)=g (): (4.b) 2

13 We select f(; ); g and g such that the eact solution is u(; ) = e + : (4.c) We perform several tests on this eample to stud the effect of boundar conditions on the qualit of our a posteriori error estimates and show the superconvergence points of the discontinuous Galerkin solution. We start b solving (4.) on a uniform mesh having 64 elements with p =; 2; 3 with U being the true boundar conditions. We repeat the previous eperiment with U being the interpolant of true boundar conditions at Radau points as defined in (2.6). We present the local effectivit indices in Figures 2 and 3. The effectivit indices corresponding to the eact boundar conditions are farther from unit than those corresponding to interpolated boundar conditions, especiall on elements near the inflow boundaries. In both eperiments the effectivit indices converge to one under p refinement. We solve (4.) using U defined in (2.6) on a 6-element uniform mesh with p ranging from to 6 and show the zero-level curves of the true error on each element in Figure 4. As predicted b the theor of 2, these results indicate that the p degree discontinuous Galerkin solution is superconvergent at Radau points, shown b `', on each element for p. Since there is no superconvergence for global errors when p =, the error estimation procedures (3.3), (3.6) and (3.7) do not appl. This is illustrated b solving (4.) and (3.4) using approimate boundar conditions on a -element mesh with p = and presenting the local effectivit indices in Figure 5. In this case we observe that the local effectivit indices get larger than unit awa from the inflow boundar elements. As a final test, we solve (4.) on uniform meshes having 25; ; 225; 4; 625 and 9 elements with p = ; 2; 3; 4 using the true boundar conditions. We present the true errors in Table and the global effectivit indices in Table 2. We repeat the previous eperiment using approimate boundar conditions (2.6) with all other parameters unchanged and present the errors and global effectivit indices in Tables 3 and 4, respectivel. The computational results indicate that the error estimates obtained using the procedure (3.3) converge to the true error under both h and p refinements. This is the first a posteriori finite element error estimate that ehibits convergence under h and p refinement for multi-dimensional problems. Since the effect of the true versus approimate boundar conditions on the effectivit indices is negligible, we shall use the approimate boundar conditions in the remainder of this section. Eample 2. We solve (4.) on the quadrilateral domain Ω = ABCD where A = (; ), B = (; :), C = (; ) and D = (; :), on a 6-element mesh for p = to 4 and show the zero-levels of the contour plot of the error on each element in Figure 6. The computational results suggest that the DG solution is superconvergent at the Radau points on more general meshes. To show the convergence of our a posteriori error estimates under mesh refinement, we solve the previous problem on meshes having 25; ; 225; 4; 625; 9 elements and p = to 4. We present the effectivit indices in Table 5 which indicate that the error estimates converge to the true errors under both h and p refinements for quadrilateral meshes. 3

14 Figure 2: Local effectivit indices for Eample on a 64-element uniform mesh and p =; 2; 3 (top to bottom) with true boundar conditions. 4

15 Figure 3: Local effectivit indices for Eample on a 64-element uniform mesh and p = ; 2; 3 (top to bottom) with approimate boundar conditions. 5

16 Figure 4: ero-level curves of the contour plot of the DG error for Eample on a 6-element uniform mesh and p = to 6 (upper left to lower right) with approimate boundar conditions. Radau points are shown with an `'. N p= p =2 p =3 p = e e e e e e e e e-3.335e e e e e e e e e e-9.686e e-4.63e e e-2 Table : jjejj L2 (Ω) for Eample on uniform meshes having 25; ; 225; 4; 625; 9 elements and p = to 4 with true boundar conditions. 6

17 Local Effectivit Inde using p = with interpolated boundar conditions Y X Figure 5: Local effectivit indices for Eample on a -element uniform mesh for p = with approimate boundar conditions. N p = p=2 p =3 p = Table 2: Effectivit indices for Eample on uniform meshes having 25; ; 225; 4; 625; 9 elements and p = to 4 with true boundar conditions. N p= p =2 p =3 p = e e e e e e e-7.654e e-3.339e e e e e-6.893e e e e e e e-4.633e e e-2 Table 3: jjejj L2(Ω) for Eample on uniform meshes having 25; ; 225; 4; 625; 9 elements and p = to 4 with approimate boundar conditions. 7

18 N p = p=2 p =3 p = Table 4: Effectivit indices for Eample on uniform meshes having 25; ; 225;4; 625; 9 elements and p = to 4 with approimate boundar conditions Figure 6: ero-level curves of the contour plot of the DG discretization error on a 6- element uniform mesh and p = to 4 (upper left to lower right) with approimate boundar conditions. Radau points are shown with an `'. 8

19 N p = p=2 p =3 p = Table 5: Global effectivit indices for Eample 2 on uniform meshes having 25; ; 225; 4; 625; 9 elements and p = to 4. Eample 3. We consider the following problem with a contact discontinuit u +2u =; (; ) 2 [; ] 2 ; (4.2a) subject to the boundar conditions u(; ) = e 2 ;»» ; (4.2b) u(;)=e + :25; <» : (4.2c) The eact solution is u(; ) = ( e 2+ + :25 if <=2 e 2+ if =2 : (4.2d) The true solution has a contact discontinuit along = 2. Therefore, the smoothness assumption of Theorem 2. is violated and as a result we epect the a posteriori error estimate to perform poorl near the discontinuit. We solve (4.2) on meshes having and 2 2 elements with p = ; 2 and present the local effectivit indices in Figure 7. These computational results indicate that the local effectivit indices on elements awa from the discontinuit converge to unit under mesh refinement. Our error estimates perform poorl on elements near the discontinuit. Since we are not using limiting to suppress spurious oscillations near the discontinuit, the region around the discontinuit where the error is underestimated gets wider as p increases. Eample 4. We consider the inviscid Burger's equation subject to the boundar conditions u + uu = f(; ); (; ) 2 [ ; ] 2 ; (4.3a) u(; ) = g (); and u(; ) = g (): (4.3b) 9

20 Figure 7: Local effectivit indices for Eample 3 with (N;p) = (24,), (4, ), (24,2) and (4,2) (upper left to lower right). 2

21 We select f, g and g such that the eact solution is u(; ) = p : (4.3c) We solve (4.3) on meshes having 35; 4; 35; 56; 875; 26; 75 rectangular elements with an element aspect ratio = = 5=7 and p = to 4, where and denote the length and width, respectivel, of an element. We compute a posteriori error estimates b solving the linear problem (3.6) and show the local effectivit indices in Table 6. We appl Newton's iteration to solve the nonlinear problem (3.7) using the linear error estimate (3.6) as an initial guess and present the effectivit indices in Table 7. While effectivit indices for both estimators are within 2% from unit, the error estimates obtained b solving the linear problem (3.6) are more efficient. For the remaining computational eamples we will present numerical results for the linear error estimator onl. N p = p =2 p = 3 p= Table 6: Effectivit indices Problem (4.3) using the linearized error estimator (3.6) on meshes having 35; 4; 35; 56; 875; 26 and 75 elements and p =to4. N p = p =2 p = 3 p= Table 7: Effectivit indices for problem (4.3) using the nonlinear error estimator (3.7) on meshes having 35; 4; 35; 56; 875; 26; 75 elements and p =to 4. Net we consider the homogeneous inviscid Burger's equation (4.3a) with f(; ) =, g (; ) = + sin(ß)=2 (4.4) and select g (;) such that the true solution is periodic and forms a shock discontinuit at the point ( 2 ß ; 2 ) which propagates along =. First, we solve this problem on ß 2

22 [ ; ] [; :4] with a smooth solution on meshes having 35; 4; 35; 56; 875; 26; 75 elements with an element aspect ratio = = 25=7 and p = to 4. We compute an error estimate b solving (3.6) and present effectivit indices in Table 8. This eample shows that the effectivit indices converge to one onl under h refinement which is due to the steepening of the wave as the shock forms. N p= p =2 p=3 p = Table 8: Effectivit indices for homogeneous Burger's equation (4.3a) with initial condition (4.4) on [ ; ] [; :4] using the linearized error estimate (3.6) on meshes having 35; 4; 35; 56; 875; 26; 68 elements and p = to 4. We conclude b solving the previous problem on [ ; ] [; :999] on meshes having N = 26 and 4 elements with an element aspect ratio = =7=5 and p =; 2. We plot the local effectivit indices in Figure 8. These computational results indicate that the theoretical results of theorem 2. are valid for nonlinear problems in regions where the solution is smooth, i.e., the local effectivit indices converge to unit under mesh refinement in regions where the solution is smooth. The regions in the lower right corner of the domain with underestimated errors correspond to regions of relativel small discretization errors and, thus, should not affect the qualit of the global effectivit inde. 5 Conclusion We showed that the discontinuous Galerkin finite element error can be split into an O(h p+ ) leading component and a higher order component. We further showed that the leading term is spanned b two (p +) degree Radau polnomials in the and directions, respectivel. We used this result to construct an a posteriori estimate of the multi-dimensional discontinuous Galerkin finite element error for hperbolic problems on rectangular meshes. The error estimation procedure is simple, can be computed in several norms and does not require an communication across neighboring elements. The later propert makes this estimation procedure useful for parallel computations. The a posteriori error estimates developed in this paper readil etend to three-dimensional heahedral elements. For a linear model problem we were able to show that the flu is strongl superconvergent on average on the outflow boundaries. This indicate that the error in the discontinuous Galerkin methods propagates at a higher order. Superconvergence at the outflow boundaries of elements for nonlinear conservation laws and 22

23 Figure 8: Local effectivit indices for the homogeneous Burger's equation (4.3a) with initial condition (4.4) on [ ; ] [; :999] using meshes having (N;p) =(26,), (4,), (26,2) and (4,2) (upper left to lower right). 23

24 problems of the form (2.44) is currentl under investigation. We plan to etend these results to locall refined meshes with hanging nodes and unstructured tetrahedral meshes for hperbolic sstems. Finall, we note that our error estimate does not appl near discontinuities. Acknowledgement Portions of this research were supported b the National Science Foundation (Grant Number DMS-7474) and Sandia National Laboratories (Contract Number AW5657). References [] S. Adjerid, K. D. Devine, J. E. Flahert, and L. Krivodonova. A posteriori error estimation for discontinuous galerkin solutions of hperbolic problems. Computer Methods in Applied Mechanics and Engineering, 9:97 2, 22. [2] F. Bassi and S. Reba. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comp. Phs., 3: , 997. [3] C. E. Baumann and J. T. Oden. A discontinuous hp finite element method for convection-diffusion problems. Computer Methods in Applied Mechanics and Engineering, 75:3 34, 999. [4] K. S. Be and J. T. Oden. hp-version discontinuous Galerkin method for hperbolic conservation laws. Computer Methods in Applied Mechanics and Engineering, 33: , 996. [5] K. S. Be, J. T. Oden, and A. Patra. hp-version discontinuous Galerkin method for hperbolic conservation laws: A parallel strateg. International Journal of Numerical Methods in Engineering, 38: , 995. [6] K. S. Be, J. T. Oden, and A. Patra. A parallel hp-adaptive discontinuous Galerkin method for hperbolic conservation laws. Applied Numerical Mathematics, 2:32 386, 996. [7] R. Biswas, K. Devine, and J. E. Flahert. Parallel adaptive finite element methods for conservation laws. Applied Numerical Mathematics, 4: , 994. [8] B. Cockburn. A simple introduction to error estimation for nonlinear hperbolic conservation laws. In Proceedings of the 998 EPSRC Summer School in Numerical Analsis, SSCM, volume 26 of the Graduate Student's Guide for Numerical Analsis, pages -46, Berlin, 999. Springer. [9] B. Cockburn and P. A. Gremaud. Error estimates for finite element methods for nonlinear conservation laws. SIAM Journal on Numerical Analsis, 33: ,

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