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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
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 $\Delta x$ only. For example, when polynomials of degree $k$ are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order $k+1/2$ in the $L^2$-norm, whereas the post-processed approximation is of order $2k+1$; if the exact solution is in $L^2$ only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order $k+1/2$ in $L^2(\Omega_0)$, where $\Omega_0$ is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.

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Additional Information

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

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

Chi-Wang Shu
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912

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

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

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