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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Error estimates in $L^{2}$, $H^{1}$ and $L^{\infty}$
in covolume methods
for elliptic and parabolic problems:
A unified approach


Authors: So-Hsiang Chou and Qian Li
Journal: Math. Comp. 69 (2000), 103-120
MSC (1991): Primary 65F10, 65N20, 65N30
Published electronically: August 25, 1999
MathSciNet review: 1680859
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Abstract: In this paper we consider covolume or finite volume element methods for variable coefficient elliptic and parabolic problems on convex smooth domains in the plane. We introduce a general approach for connecting these methods with finite element method analysis. This unified approach is used to prove known convergence results in the $H^1, L^2$ norms and new results in the max-norm. For the elliptic problems we demonstrate that the error $u-u_h$ between the exact solution $u$ and the approximate solution $u_h$ in the maximum norm is $O(h^2|\ln h|)$ in the linear element case. Furthermore, the maximum norm error in the gradient is shown to be of first order. Similar results hold for the parabolic problems.


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Additional Information

So-Hsiang Chou
Affiliation: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403-0221, U.S.A.
Email: chou@zeus.bgsu.edu; http://www-math.bgsu.edu/~chou

Qian Li
Affiliation: Department of Mathematics, Shandong Normal University, Shandong, China

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01192-8
PII: S 0025-5718(99)01192-8
Keywords: Covolume methods, finite volume methods, generalized difference methods, network methods, finite volume element
Received by editor(s): March 19, 1996
Received by editor(s) in revised form: April 22, 1996
Published electronically: August 25, 1999
Article copyright: © Copyright 1999 American Mathematical Society