computations. Furthermore, they ehibit strong superconvergence of DG solutions and flues for hyperbolic [,, 3] and elliptic [] problems. Adjerid et al

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1 Superconvergence of Discontinuous Finite Element Solutions for Transient Convection-diffusion Problems Slimane Adjerid Department of Mathematics Virginia Polytechnic Institute and State University Blacksburg, VA 46 Andreas Klauser Technical University of Munich Boltzmann Str. 3, Garching Munich, Germany June 6, 4 Abstract We present a study of the local discontinuous Galerkin method for transient convection-diffusion problems in one dimension. We show that p degree piecewise polynomial discontinuous finite element solutions of convection-dominated problems are O( p+ ) superconvergent at Radau points. For diffusion-dominated problems, the solution's derivative iso( p+ ) superconvergent at the roots of the derivative of Radau polynomial of degree p +. Using these results, we construct several asymptotically eact a posteriori finite element error estimates. Computational results reveal that the error estimates are asymptotically eact. Introduction Discontinuous Galerkin (DG) methods have attracted much attention in the past ten years. The DG method was first used for the neutron equation []. Since then, DG methods have been used to solve hyperbolic [6, 7, 8, 9, 4, 5, 7, ], parabolic [8, 9], and elliptic [4, 5, 3] partial differential equations. For a more complete list of citations on the DG method and its applications consult [3]. Since no continuity across element boundaries is needed, the shape functions on each element can be selected to be orthogonal which leads to diagonal mass matrices and well conditioned algebraic systems for higher-order finite element approimations. Moreover, DG methods are well suited to solve problems on locally refined meshes with hanging nodes. They have a very simple communication pattern between elements which makes them ideal for parallel

2 computations. Furthermore, they ehibit strong superconvergence of DG solutions and flues for hyperbolic [,, 3] and elliptic [] problems. Adjerid et al. [] showed that the DG solution of one-dimensional problems is O( p+ ) superconvergent at the downwind end of each element. At the other Radau points the solution is O( p+ ) superconvergent. These results were etended to twodimensional problems on rectangular meshes [] where the DG solution is O( p+ ) at the Radau points. On average, the flu on the outflow boundary of the first inflow element is O( p+ ) superconvergent. Similar results hold for nonlinear hyperbolic problems [3]. Castillo [] investigated the superconvergence behavior of the LDG method applied to a two-point elliptic boundary-value problem using the numerical flu proposed in []. He showed that on each element the p degree LDG solution gradient iso( p+ ) superconvergent at the shifted roots of the p degree Legendre polynomial. In this manuscript we present anumerical study of superconvergence properties for the Local Discontinuous Galerkin (LDG) method [6] applied to time dependent convectiondiffusion problems. In particular, we show that LDG solutions of convection-dominated problems are O( p+ ) superconvergent at the shifted Radau points on each element. For diffusion-dominated problems, the derivative of the LDG solution is O( p+ ) superconvergent at the roots of the derivative of Radau polynomial of degree p +. This study indicates that superconvergence of LDG solutions depends strongly on the numerical flu and that some numerical flues yield richer superconvergence behavior than others. This manuscript is organized as follows: In we present a model problem and recall the LDG formulation. In 3 we present numerical results for linear and nonlinear convection-dominated problems. We study diffusion and diffusion-dominated problems in 4. We conclude with a discussion of our findings in 5. The Local Discontinuous Galerkin method The Local Discontinuous Galerkin method for convection-diffusion problems was introduced by Cockburn in [6]. Several a priori error estimates have been established for linear problems [,, 6]. Here, we consider the scalar convection-diffusion problem u t +(f(u) du ) = g; a<<b; t>; (.a) subject to the initial and boundary conditions u(; ) = u (); a<<b; (.b) u(a; t) =u a (t); u(b; t) =u b (t); t>: (.c) In our analysis we assume that f (u) > ; d and select f(u), g(; t), u (), u a (t), u b (t) such that the eact solution, u, is smooth. Following [], we introduce the auiliary variable q = p du ; (.a)

3 p p h =(h u ;h q ) T =(f(u) dq; du); (.b) and write (.) as u t +(h u ) = g q +(h q ) = a<<b; t>; (.c) (.d) subject to the initial and boundary conditions (.b) and (.c). Let us partition [a; b] into N subintervals I j = [ j ; j ]; j = ; ; N, with j = j j and =(b a)=n. The LDG weak formulation is obtained by multiplying (.c) and (.d) by a test function (v; r) and integrating by parts to obtain (u t ;v) j (h u ;v ) j + h u v j j + j (q; r) j (h q ;r ) j + h q r j j + j = (g; v) j (.3a) = ; (.3b) for all v; r H ; where the left and right limits are defined as w( i ) = lim w(), respectively.! i ;> i The element inner product is defined as Z j lim w() and w( + ) = i! i ;< i (u; v) j = j uvd: We consider the L norm of square integrable functions as kuk L = Z b a u d : (.4a) Let H s be the Sobolev space of square integrable functions with all k u; k = ; ; ::; s being square integrable and equipped with the norm kuk s = sx fl k= k ufl fl L ; s =; ;::: : (.4b) of discontinuous piecewise polynomial func- We construct a finite dimensional space V p N tions such that where P p denotes the space of polynomials of degree p. V p N = fv j V j I j P p g; (.5) 3

4 The discrete LDG formulation consists of finding U and Q V p N such that (U t ;V) j (h U ;V ) j + ^h U V j j + j (Q; R) j (h Q ;R ) j + ^h Q R j j + j = (g; V ) j (.6a) = ; (.6b) for all V; R V p N ; subject to the initial condition U(; ) V p N obtained by interpolating u on each interval at the shifted roots of the right Radau polynomial R + p+(ο) =P p+ (ο) P p (ο);» ο» ; (.7) where P p (ο) is the Legendre polynomial of degree p. The left Radau polynomial of degree p + is defined as R p+(ο) =P p+ (ο)+p p (ο);» ο» : (.8) Since the trial function is discontinuous, the flues at the end points are replaced by the following numerical flues where ^h( j )=(c μ U;) T p d( μ Q; μ U) T c c c [U] [Q] ; (.9a) [u] =u + u and μu =(u + + u )=: (.9b) The numerical flu at the boundary points is well defined by setting (u; q)(a ;t) = (u a (t);q(a + ;t)) (.9c) (u; q)(b + ;t) = (u b (t);q(b ;t)): (.9d) To obtain the optimal hp method [] we use ρ c= for j =;::: ;N c = mafc=; maf;p N gd= N g for j = N ff ; (.9e) c = p d=: (.9f) On each element the trial and test functions are represented as a linear combination of Legendre polynomials which leads to diagonal mass matrices and well-conditioned systems. For piecewise smooth functions we use the broken Sobolev norm kuk s;b = N X j= jjujj s;j (.) where the subscript j indicates that integrals are restricted to the interval [ j ; j ]. 4

5 3 Error estimation for convection-diffusion problems We consider the linear convection-diffusion problem (.) where f(u) =cu; c>; (3.a) and select the initial and boundary conditions such that the eact solution is u(; t) =e dßt sin ß( ct);»» ;» t» : (3.b) We set c =, d = 4 and solve (., 3.) on uniform meshes having 8; 6; 3 and 64 elements with p = ; ; ; 3; 4. Time integration is performed using the Matlab routine ode45 with AbsTol = :3 4 and RelTol = :3 4. We show plots of the true error (u U)(; ) for p = ; ; ; 3; 4 and N = 6 in Figure. The global error is superconvergent at the shifted roots of Radau polynomial R + p+ ; p. For p = the global error is not superconvergent at the left downwind end of the elements which is in agreement with the results of Adjerid et al. []. These numerical results also show that the roots of the true error get closer to the roots of Radau polynomials with increasing p. In Figure we plot, in log-log scale, the maimum error at the shifted Radau points ku Uk ;Λ (t) = ma ma j(u U)( j;i ;t)j; (3.)»j»N»i»p+ where j;i ;i=; ; ;p+ are the shifted roots of R + p+ on the interval [ j ; j ]. These results suggest that the LDG solution converges at an O( p+ ) rate at Radau points while globally it converges at an O( p+ ) rate. Now, we solve the previous problem for p =; ; ; 3; 4 on the 6-element nonuniform mesh! ψe j 6 j = e :5 ; j =; ; 6: (3.3) We plot the error (u U)(; ) in Figure 3. Again, we observe that the LDG solution is superconvergent at the shifted Radau points. The previous numerical results suggest that the leading term in the LDG finite element error on each element is proportional to Radau polynomial R + p+(). Thus, we approimate the finite element error as u U ß E u = ff j (t)r + p+(); (3.4a) q Q ß E q = fi j (t)r + p+(); j << j ; t>: (3.4b) Computational results indicate that the leading term in q Q is not proportional to the right Radau polynomial. However, the assumption (3.4b) does not affect the quality of the a posteriori error estimate for u. Setting u ß U + ff j R + p+ and q ß Q + fi j R + p+ in (.3) and testing against R + p+ to obtain a weak formulation for the error which consists of determining ff j (t) and fi j (t) such that 5

6 u U u U u U u U u U Figure : The true error (u U)(; ) for problem (.,3.) on a 6 element uniform mesh and p =; ; ; 3; 4 (upper left to lower center). Shifted Radau points are marked by +. 6

7 p= p= p= p=3 p=4 T 4 T 3 u U,* () 6 8 T 6 T 4 T 5 / Figure : Maimum error ku Uk ;Λ () for problem (.,3.) using p = ; ; 3; 4. Triangle T m has a slope m. ((U + E u ) t ;R + p+) j (h u (U + E u ;Q+ E q );R + p+) j + ^h u (U + E u ;Q+ E q )R + p+ j j + j =(g; R + p+) j ; (3.5a) (Q + E q ;R + p+) j (h q (U + E u ;Q+ E q );R + p+) j + ^h q (U + E u ;Q+ E q )R + p+ j j + j =: (3.5b) We note that on each element we have a set of ordinary differential equations for ff j (t) and fi j (t). The initial values ff j (); j =; ; ;N;are obtained using the L projection (U(:; ) + E u (:; );R p+ (:))=(u (:);R p+ (:)): (3.6) The initial values fi j (); j =; ; ;N; are obtained from (3.5b). Since the leading term of the error on each element converges to a Radau polynomial with increasing t, we improve the quality of the error estimator for t by interpolating the initial condition u () at the shifted roots of R + p+. Now, we solve (.,3.) using a uniform mesh having N = 6 elements with p = and integrate from t = to t = again by the Matlab routine ode45 with the same 7

8 u U u U u U u U u U Figure 3: The true error (u U)(; ) for problem (.,3.) on the 6 element mesh (3.3) and p =; ; ; 3; 4 (upper left to lower center). Shifted Radau points are marked by +. 8

9 prescribed tolerances as mentioned above. We plot the true error, (u U)(; ), and the error estimate, E u (; ), versus in Figure 4 and observe that the estimated error stays very close to the true error u U and E u Figure 4: The true (u U)(; ) (solid) and E u (; ) (dashed) for problem (.,3.) at t =with N = 6 and p =. Radau points are marked by +. An accepted measure of the quality of a posteriori error estimates is the effectivity inde in H s and L norms defined as H s(t) = jje u(:; t)jj s;b jj(u U)(:; t)jj s;b ; (3.7a) L (t) = jje u(:; t)jj L : (3.7b) jj(u U)(:; t)jj L Ideally, the effectivity indices should stay close to one and should converge to one under mesh refinement. We solve (.,3.) with d = 4 on uniform meshes having N = 8 and N = 6 and p =; ; 3; 4. In Figure 5 and 6, respectively, we plot H (t) versus time for N =8and N = 6. Effectivity indices stay close to unity for all times and converge under h and p refinements. In order to study the effect of the diffusion term, we solve (., 3.) with d = 4 ; ; :, using uniform meshes having N = 8; 6; 3 elements and p = ; ; 3; 4. 9

10 . p= p= p=3 p=4.5 θ H (t) t Figure 5: Effectivity indices H (t) versus time for problem (.,3.) using N = 8 and p =; ; 3; 4. d = 4 d = d =: p N =8 N =6 N =3 N =8 N =6 N =3 N =8 N =6 N = Table : Effectivity indices H () for problem (., 3.) with d = 4, d = and d = using N =8; 6; 3 and p =; ; 3; 4.

11 . p= p= p=3 p=4. θ H (t) t Figure 6: Effectivity indices H (t) versus time for problem (.,3.) using N = 6 and p =; ; 3; 4. We use the Matlab routine ode45 with AbsTol = and RelTol = to integrate from t =tot = and show the effectivity indices H () in Table. We note that the effectivity indices stay close to unity for a wide range of mesh sizes and polynomial degrees. Numerical results further indicate that the error estimates converge to the true error with decreasing mesh size and increasing polynomial degree p. For hyperbolic problems, error estimates in the L norm are more appropriate since solution can develop shocks. Net, we solve (.,3.) with d = :; :; :5 using uniform meshes having N =8; 6; 3 elements and p =; ; 3; 4 and present the effectivity indices at t = in the L norm in Table. d =: d =: d =:5 p N =8 N =6 N =3 N =8 N =6 N =3 N =8 N =6 N = Table : Effectivity indices L () for problem (.,3.) with d = :, d = : and d =:5 using N =8; 6; 3 and p =; ; 3; 4. The results presented in Table indicate that the quality of L (t) based on the

12 assumption (3.4) deteriorates with increasing d. In 4, we will study a pure diffusion problem and design a more robust error estimate for diffusion-dominated problems. Net, we consider the viscous Burgers' equation with f(u) = u written as u t + uu du = g(; t); <<; <t<: (3.8a) We select g, the initial and boundary conditions such that eact solution is u(; t) =e + +t : (3.8b) Let us use the numerical flu where ^h( j )=((f(u + )+f(u ))=; ) T p d(μq; μu) T c c c [u] [q] ; (3.9a) c = p d=; c = ma j f (u) j= u[u ;u + ] mafj u j; j u + jg: (3.9b) At the boundary points we use the flu (f(u);q)(a ;t) = (f(u a (t));q(a + ;t)) (3.) (f(u);q)(b + ;t) = (f(u b (t));q(b ;t)): (3.) We select d = 4 and solve (3.8) using uniform meshes having N = 8; 6 and 3 elements with p = ; ; 3; 4 and integrate from t = to t = using the Matlab routine ode45 with AbsTol = and RelTol =. We present the effectivity indices at t = in the H and L norms in Table 3. We observe that the error estimates converge to the true error in the H and L norms under h and p refinements. L () H () p N =8 N =6 N =3 N =8 N =6 N = Table 3: Effectivity indices L () and H () for problem (3.8) with d = 4 on uniform meshes having N =8; 6; 3 elements with p =; ; 3; 4. 4 Error estimation for diffusion-dominated problems Here we consider the linear diffusion problem u t = u ; <<; t>; (4.a)

13 and select the initial and boundary conditions such that the eact solution is u(; t) =e ßt sin ß: (4.b) We solve (.6) and (.9) on a uniform mesh having N = 8 elements with p = ; 3 and integrate on» t» :5 using the Matlab routine ode45 with AbsTol = 3 and RelTol =. We plot the errors u U and q Q versus at t = :5 in Figure 7. We observe that u and q do not ehibit any superconvergence. However, u U and q Q, shown in Figure 8, indicate that on each element U is superconvergent at the shifted roots of R p+() + while Q is superconvergent at the shifted roots R p+(). From several numerical eperiments we conclude that on each element we have u U ß E u (; t) =ff j (t)r + p+()+c j (t); (4.a) q Q ß E q (; t) =fi j (t)r p+()+z j (t); j»» j ; t : (4.b) u U q Q u U 4 q Q Figure 7: True errors (u U)(; :5) (upper left) and (q Q)(; :5) (lower left) for problem (4.) with N = 8 and p =. The error u U (upper right) and q Q (lower right) for N =8and p =3. Net, we solve (.6) and (.9) on uniform meshes having N = 4; 8; 6; 3 elements with p = ; ; 3; 4 and integrate on» t» :5 using ode45 with AbsTol = 6 and RelTol = :3 4. In Figure 9 we plot the maimum error (u U )(; :5) and (q Q )(; :5) at the roots of R p+() + and R p+() versus =. These results show that U and Q are O( p+ ) superconvergent at the roots of R p+() + and R p+(), 3

14 u U.5.5 q Q u U q Q Figure 8: True errors (u U )(; :5) (top left) and (q Q )(; :5) (top right) for problem 4. with N = 8 and p =. The error (u U )(; :5) (bottom left) and (q Q )(; ) (bottom right) for N = 8 and p =3. The shifted roots of R + p+ (left) and R p+ (right) are marked by +. 4

15 respectively. We define k(u U) (:; t)k ;+ (t) = ma»j»n ma»i»p j(u U) ( + j;i ;t)j (4.3a) and k(q Q) (:; t)k ; (t) = ma»j»n ma»i»p j(q Q) ( j;i ;t)j (4.3b) where ± j;i denote the shifted roots of R ± p+ on the interval [ j ; j ]. In order to estimate the error we use (4.) to approimate u Uand q Q by E u and E q, respectively, and solve the problem ((U + E u ) t ;w ) j (h u (U + E u ;Q+ E q );w ) j + ^h u (U + E u ;Q+ E q )w j j + j =(g; w ) j ; (4.4a) (Q + E q ;w ) j (h q (U + E u ;Q+ E q );w ) j + ^h q (U + E u ;Q+ E q )w j j + j = (4.4b) for w =;R + p+ ; w =;R p+ : The initial conditions ff j () and c j () are obtained using the L projection (U(:; ) + E u (:; );V)=(u ;V); for V =;R + p+(): (4.5) Initial values for fi j and z j are obtained from (4.4b). This leads to a system of ordinary differential equations for ff j (t) and fi j (t) which is integrated using the Matlab routine ode45 with AbsTol = and RelTol =. L (:5) H (:5) H (:5) p N =8 N =6 N =3 N =8 N =6 N =3 N =8 N =6 N = Table 4: Effectivity indices L (:5), H (:5) and H (:5) for problem (4.) using N = 8; 6; 3 with p =; ; 3. We solve (4.) on uniform meshes having N = 8; 6; 3 element with p = ; ; 3; 4 and plot the effectivity indices H (t), for N=8, versus t in Figure. The effectivity indices L (:5), H (:5) and H (:5), shown in Table 4, suggest that the effectivity indices converge to unity under h refinement. Error estimates for u are also available 5

16 p= p= p=3 p=4 3 4 T 3 u U,+ (.5) T 5 8 T 5 9 T 7 / p= p= p=3 p=4 q Q, (.5) 4 6 T 3 T 5 8 T 7 T 5 / Figure 9: Maimum error jj(u U )jj ;+ (:5) (top) and maimum error of jjq Q )jj ; (:5) (bottom) for problem (.) using p = ; ; 3; 4. Triangle T m has slope m. 6

17 .5 8 Plot of u u h (red if known) and error estimate (optional,green).5 (u U)(,.5), E n (,.5) Figure : The true error (u U)(; :5) (solid) and the error estimate E u (; :5) (dashed) for problem (4.) using N = 3 and p = p= p= p=3 p=4 Effectivity inde in H norm t Figure : Effectivity indices H (t) versus t for problem (4.) using p = ; ; 3; 4 and N =8. 7

18 through q = p du. The errors shown in Figure demonstrate that the estimate E u stay close to the true error u U over the whole domain. Thus, our error estimates provide an accurate measure of the true error in the L, H, H and infinity norms. In our last eample we solve the nonlinear viscous Burgers' equation (3.8) with d = on uniform meshes having N =8; 6; 3 elements and p =; ; 3. The effectivity indices L (:5) and H (:5), shown in Table 5, indicate that our error estimate is a good approimation to the true error in L and H norms. Finally, the effectivity indices shown in Table 5 seem to converge to unity under h and p refinements. L (:5) H (:5) p N =8 N =6 N =3 N =8 N =6 N = Table 5: Effectivity indices H (:5) and L (:5) for problem (3.8) with d = using N =8; 6; 3 elements and p =; ; 3. 5 Conclusion Numerical evidence suggests that the superconvergence properties of discontinuous solutions of one-dimensional hyperbolic problems etend to LDG solutions of transient convection-dominated problems. Thus, the LDG solution of degree p is O( p+ ) superconvergent at the roots of (p + )-degree Radau polynomials. We used these results to construct aposteriori error estimates that converge to the true error u U under h and p refinements. On the other hand, LDG solutions of diffusion and diffusion-dominated problems ehibit an O( p+ ) superconvergence rate for U and Q at the roots of R p+() + and R p+(), respectively. Again, we used these superconvergence results to compute efficient and asymptotically eact aposteriori error estimates in L, H and H norms. The results shown here and in [], indicate that the superconvergence behavior of LDG solutions depends strongly on the numerical flu. We are currently investigating superconvergence of LDG methods for other eisting numerical flues. The present error analysis can be easily etended to two-dimensional problems on rectangular meshes by following the same line of reasoning as in []. Error estimates on triangular meshes may also be obtained following []. Additional work will be necessary to establish rigorous proofs of these results. Acknowledgement Portions of this research were supported by the National Science Foundation (Grant Number DMS-7474) and Sandia National Laboratories (Contract Number AW5657). 8

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