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

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- by Bernardo Cockburn, Mitchell Luskin, Chi-Wang Shu and Endre Süli;
- Math. Comp.
**72**(2003), 577-606 - DOI: https://doi.org/10.1090/S0025-5718-02-01464-3
- Published electronically: November 20, 2002
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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.## References

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## Bibliographic 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
- MR Author ID: 242268
- 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
- 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 - © Copyright 2002 American Mathematical Society
- Journal: Math. Comp.
**72**(2003), 577-606 - MSC (2000): Primary 65M60, 65N30, 35L65
- DOI: https://doi.org/10.1090/S0025-5718-02-01464-3
- MathSciNet review: 1954957