TY - JOUR

T1 - Enhanced accuracy by post-processing for finite element methods for hyperbolic equations

AU - Cockburn, Bernardo

AU - Luskin, Mitchell

AU - Shu, Cht Wang

AU - Süli, Endre

N1 - Copyright:
Copyright 2004 Elsevier Science B.V., Amsterdam. All rights reserved.

PY - 2003/4

Y1 - 2003/4

N2 - 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 Δx 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 L2-norm, whereas the post-processed approximation is of order 2k + 1; if the exact solution is in L2 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 L2 (Ω0), where Ω0 is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.

AB - 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 Δx 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 L2-norm, whereas the post-processed approximation is of order 2k + 1; if the exact solution is in L2 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 L2 (Ω0), where Ω0 is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.

KW - Finite element methods

KW - Hyperbolic problems

KW - Post-processing

UR - http://www.scopus.com/inward/record.url?scp=0037376067&partnerID=8YFLogxK

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U2 - 10.1090/S0025-5718-02-01464-3

DO - 10.1090/S0025-5718-02-01464-3

M3 - Article

AN - SCOPUS:0037376067

VL - 72

SP - 577

EP - 606

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 242

ER -