Enhanced accuracy by post-processing for finite element methods for hyperbolic equations

Mathematics of Computation

Volumn 72, Issue 242, 2003, Pages 577-606

  Cockburn, Bernardo ^a   Luskin, Mitchell ^a   Shu, Cht Wang ^b   Süli, Endre ^c  

^a UNIVERSITY OF MINNESOTA   (United States)

^b BROWN UNIVERSITY   (United States)

^c UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Finite element methods; Hyperbolic problems; Post processing

Indexed keywords

EID: 0037376067     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-02-01464-3     Document Type: Article

References (24)

1. S. Adjerid, M. Aiffa, and J. E. Flaherty, Computational methods for singularly perturbed systems, Singular Perturbation Concepts of Differential Equations (J. Cronin and R.E. O'Malley, eds.), AMS Proceedings of Symposia in Applied Mathematics, AMS, 1998. MR 2001c:65097
2. S. Adjerid, M. Aiffa, and J.E. Flaherty, High-order finite element methods for singularly perturbed elliptic and parabolic problems, SIAM J. Appl. Math. 55 (1995), 520-543. MR 96d:65182
3. M.Y.T. Apelkrans, Some properties of difference schemes for hyperbolic equations with discontinuities and a new method with almost quadratic convergence, Tech. Report 15A, Uppsala University, Dept. of Computer Science, 1969.
4. L.A. Bales, Some remarks on post-processing and negative norm estimates for approximations to nonsmooth solutions of hyperbolic equations, Comm. Numer. Methods Engrg. 9 (1993), 701-710. CMP 93:17
5. J.H. Bramble and A.H. Schatz, Higher order local accuracy by averaging in the finite element method, Math. Comp. 31 (1977), 94-111. MR 55:4739
6. P. Brenner, V. Thomée, and L.B. Wahlbin, Besov spaces and applications to difference methods for initial value problems, Lecture Notes in Mathematics, vol. 434, Springer Verlag, 1975. MR 57:1106
7. B. Cockburn, Discontinuous Galerkin methods for convection-dominated problems, High-Order Methods for Computational Physics (T. Barth and H. Deconink, eds.), Lecture Notes in Computational Science and Engineering, vol. 9, Springer Verlag, 1999, pp. 69-224. MR 2000f: 76096
8. B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: General framework, Math, Comp. 52 (1989), 411-435. MR 90k:65160
9. _, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998), 2440-2463. MR 99j:65163
10. J. Douglas, Jr., Superconvergence in the pressure in the simulation of miscible displacement, SIAM J. Numer. Anal. 22 (1985), 962-969. MR 86j:65129
11. B. Engquist and B. Sjögreen, The convergence rate affinite difference schemes in the presence of shocks, STAM J. Numer. Anal. 35 (1998), 2464-2485. MR 99k:65080
12. R.P. Fedorenko, The application of high-accuracy difference schemes to the numerical solution of hyperbolic equations, Zh. Vychisl. Mat. i Mat. Fiz. 2 (1962), 1122-1128, (in Russian). MR 26:5739
13. D. Gottlieb and E. Tadmor, Recovering pointwise values of discontinuous data within spectral accuracy, Proceedings of U.S.-Israel Workshop, Progress in Scientific Computing, vol. 6, Birkhäuser Boston Inc., 1985, pp. 357-375. MR 90a:65041
14. B. Gustafsson, H.-O. Kreiss, and J. Ollger, Time dependent problems and difference methods, John Wiley & Sons, New York, 1995. MR 9Tc:65145
15. C. Johnson and U. Nävert, Analysis of some finite element methods for advection-diffusion problems, Mathematical and Numerical Approaches to Asymptotic Problems in Analysis (University of Nijmegen, The Netherlands, June 9-13, 1980) (L.S. Frank O. Axelsson and A. van der Sluis, eds.), North Holland Math. Stud., 47, North-Holland, Amsterdam, 1981, pp. 99-116. MR 82e:65127
16. C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp. 46 (1986). 1-26. MR 88b:65109
17. B.S. Jovanović, L.D. Iovanović, and E. Süli, Convergence of a finite difference scheme for second-order hyperbolic equations with variable coefficients, IMA Journal of Numerical Analysis 7 (1987), 39-45. MR 90a:65218
18. R.B. Lowrie, Compact higher-order numerical methods for hyperbolic conservation laws, Ph.D. thesis, University of Michigan, 1996.
19. A. Majda, J. McDonough, and S. Osher, The Fourier method for nonsmooth initial data, Math. Comp. 32 (1978), 1041-1081. MR 80a:65197
20. A. Majda and S. Osher, Propagation of error into regions of smoothness for accurate difference approximate solutions to hyperbolic equations, Comm. Pure Appl. Math. 30 (1977), 671-705. MR 57:11080
21. M.S. Mock and P.D. Lax, The computation of discontinuous solutions of linear hyperbolic equations, Comm. Pure Appl. Math. 31 (1978). 423-430. MR 57:8054
22. V. Thomée, High order local approximations to derivatives in the finite element method, Math. Comp. 31 (1977), 652-660. MR 55:11572
23. _, Negative norm estimates and superconvergence in Calerkin methods for parabolic problems, Math. Comp. 31 (1980), 93-113. MR 81a:65092
24. L.B. Wahlbin, Superconvergence in Calerkin finite element methods, Lecture Notes in Mathematics, vol. 1605, Springer Verlag, 1995. MR 98j:65083