Error estimates in , and in covolume methods for elliptic and parabolic problems: A unified approach
Authors:
SoHsiang Chou and Qian Li
Journal:
Math. Comp. 69 (2000), 103120
MSC (1991):
Primary 65F10, 65N20, 65N30
Published electronically:
August 25, 1999
MathSciNet review:
1680859
Fulltext PDF Free Access
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References 
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Additional Information
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 norms and new results in the maxnorm. For the elliptic problems we demonstrate that the error between the exact solution and the approximate solution in the maximum norm is 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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 1.
 R. E. Bank and D. J. Rose, Some error estimates for the box method, SIAM J. Numer. Anal, 24, (1987), 777787. MR 88j:65235
 2.
 S. Brenner and R. Scott, The mathematical theory of finite element methods, SpringerVerlag, New York, (1994). MR 95f:65001
 3.
 Z. Cai and S. McCormick, On the accuracy of the finite volume element method for diffusion equations on composite grids, SIAM J. Numer. Anal, 27, No. 3, (1990), 636656. MR 91d:65182
 4.
 Z. Cai, J. Mandel, and S. McCormick, The finite volume element method for diffusion equations on general triangulations, SIAM J. Numer. Anal, 28, No. 2, (1991), 392403. MR 92j:65165
 5.
 S. H. Chou, A network model for incompressible twofluid flow and its numerical solution, Numer. Meth. Partial Diff. Eqns, 5, (1989), 124. MR 90i:76142
 6.
 S. H. Chou, Analysis and convergence of a covolume method for the generalized Stokes problem, Math. Comp. 66, (1997), 85104. MR 97e:65109
 7.
 S. H. Chou and D. Y. Kwak, Mixed covolume methods on rectangular grids for elliptic problems, SIAM J. Numer. Anal, to appear.
 8.
 S. H. Chou, D. Y. Kwak and P.S. Vassilevski, Mixed covolume methods for elliptic problems on triangular grids, SIAM J. Numer. Anal., 35, No. 5, 18501861, (1998). CMP 98:17
 9.
 S. H. Chou, D. Y. Kwak and P.S. Vassilevski, Mixed upwinding covolume methods on rectangular grids for convectiondiffusion problems, SIAM J. Sci. Comput., to appear.
 10.
 S. H. Chou and P. S. Vassilevski, A general mixed covolume framework for constructing conservative schemes for elliptic problems, Math. Comp. 68, 9911011, (1999). CMP 99:11
 11.
 S. H. Chou and D. Y. Kwak, A covolume method based on rotated bilinears for the generalized Stokes problem, SIAM J. Numer. Anal., 35, No. 2, (1998), 497507. MR 99d:65302
 12.
 S. H. Chou and D. Y. Kwak, Analysis and convergence of a MAClike scheme for the generalized Stokes Problem, Numer. Meth. Partial Diff. Eqns, 13, (1997), 147162. MR 98a:65154
 13.
 C. A. Hall, J. C. Cavendish and W. H. Frey, The dual variable method for solving fluid flow difference equations on Delaunay triangulations, Comput. & Fluids, 20, No. 2, (1991), 145164. MR 92g:76059
 14.
 C. A. Hall, T. A. Porsching and P. Hu, Covolumedual variable method for thermally expandable flow on unstructured triangular grids, 2, Comp. Fluid Dyn, (1994), 111139.
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 R. H. Li and Z. Y. Chen, The Generalized difference method for differential equations, Jilin University Publishing House, (1994). (In Chinese)
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 R. H. Li and P. Q. Zhu, Generalized difference methods for second order elliptic partial differential equations (I), A Journal of Chinese Universities, (1982), 140152.
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 R. A. Nicolaides, T. A. Porsching and C. A. Hall, Covolume methods in computational fluid dynamics, in Computational Fluid Dynamics Review, M. Hafez and K. Oshma ed., John Wiley and Sons, (1995), 279299.
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Additional Information
SoHsiang Chou
Affiliation:
Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 434030221, U.S.A.
Email:
chou@zeus.bgsu.edu; http://wwwmath.bgsu.edu/~chou
Qian Li
Affiliation:
Department of Mathematics, Shandong Normal University, Shandong, China
DOI:
http://dx.doi.org/10.1090/S0025571899011928
PII:
S 00255718(99)011928
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
