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

Authors:
Bernardo Cockburn, Mitchell Luskin, Chi-Wang Shu and Endre Süli

Journal:
Math. Comp. **72** (2003), 577-606

MSC (2000):
Primary 65M60, 65N30, 35L65

DOI:
https://doi.org/10.1090/S0025-5718-02-01464-3

Published electronically:
November 20, 2002

MathSciNet review:
1954957

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Abstract: We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of only. For example, when polynomials of degree are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order in the -norm, whereas the post-processed approximation is of order ; if the exact solution is in only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order in , where is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.

**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 of finite difference schemes in the presence of shocks*, SIAM 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. Oliger,*Time dependent problems and difference methods*, John Wiley & Sons, New York, 1995. MR**97c: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. Jovanovic, L.D. Iovanovic, 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 Galerkin methods for parabolic problems*, Math. Comp.**31**(1980), 93-113. MR**81a:65092****24.**L.B. Wahlbin,*Superconvergence in Galerkin finite element methods*, Lecture Notes in Mathematics, vol. 1605, Springer Verlag, 1995. MR**98j:65083**

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Additional Information

**Bernardo Cockburn**

Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Email:
cockburn@math.umn.edu

**Mitchell Luskin**

Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Email:
luskin@math.umn.edu

**Chi-Wang Shu**

Affiliation:
Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912

Email:
shu@cfm.brown.edu

**Endre Süli**

Affiliation:
Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom

Email:
Endre.Suli@comlab.ox.ac.uk

DOI:
https://doi.org/10.1090/S0025-5718-02-01464-3

Keywords:
Post-processing,
finite element methods,
hyperbolic problems

Received by editor(s):
November 14, 2000

Published electronically:
November 20, 2002

Additional Notes:
The first author was supported in part by NSF Grant DMS-9807491 and by the University of Minnesota Supercomputing Institute

The second author was supported in part by NSF Grant DMS 95-05077, by AFOSR Grant F49620-98-1-0433, by ARO Grant DAAG55-98-1-0335, by the Institute for Mathematics and its Applications, and by the Minnesota Supercomputing Institute

The third author was supported in part by ARO Grant DAAG55-97-1-0318 and DAAD19-00-1-0405, NSF Grant DMS-9804985, NASA Langley Grant NCC1-01035 and and Contract NAS1-97046 while this author was in residence at ICASE, NASA Langley Research Center, and by AFOSR Grant F49620-99-1-0077

The fourth author is grateful to the Institute for Mathematics and Its Applications at the University of Minnesota and the University of Minnesota Supercomputing Institute for their generous support

Article copyright:
© Copyright 2002
American Mathematical Society