Enhanced accuracy by postprocessing for finite element methods for hyperbolic equations
Authors:
Bernardo Cockburn, Mitchell Luskin, ChiWang Shu and Endre Süli
Journal:
Math. Comp. 72 (2003), 577606
MSC (2000):
Primary 65M60, 65N30, 35L65
Published electronically:
November 20, 2002
MathSciNet review:
1954957
Fulltext PDF Free Access
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Abstract: We consider the enhancement of accuracy, by means of a simple postprocessing technique, for finite element approximations to transient hyperbolic equations. The postprocessing 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 postprocessed 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 postprocessed 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.
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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
ChiWang 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:
http://dx.doi.org/10.1090/S0025571802014643
PII:
S 00255718(02)014643
Keywords:
Postprocessing,
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 DMS9807491 and by the University of Minnesota Supercomputing Institute
The second author was supported in part by NSF Grant DMS 9505077, by AFOSR Grant F496209810433, by ARO Grant DAAG559810335, by the Institute for Mathematics and its Applications, and by the Minnesota Supercomputing Institute
The third author was supported in part by ARO Grant DAAG559710318 and DAAD190010405, NSF Grant DMS9804985, NASA Langley Grant NCC101035 and and Contract NAS197046 while this author was in residence at ICASE, NASA Langley Research Center, and by AFOSR Grant F496209910077
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
