Error estimates in , and

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

Full-text PDF Free Access

Abstract | References | Similar Articles | 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 max-norm. 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.

**1.**Randolph E. Bank and Donald J. Rose,*Some error estimates for the box method*, SIAM J. Numer. Anal.**24**(1987), no. 4, 777–787. MR**899703**, 10.1137/0724050**2.**Susanne C. Brenner and L. Ridgway Scott,*The mathematical theory of finite element methods*, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR**1278258****3.**Zhi Qiang Cai and Steve McCormick,*On the accuracy of the finite volume element method for diffusion equations on composite grids*, SIAM J. Numer. Anal.**27**(1990), no. 3, 636–655. MR**1041256**, 10.1137/0727039**4.**Zhi Qiang Cai, Jan Mandel, and Steve McCormick,*The finite volume element method for diffusion equations on general triangulations*, SIAM J. Numer. Anal.**28**(1991), no. 2, 392–402. MR**1087511**, 10.1137/0728022**5.**S. H. Chou,*A network model for incompressible two-fluid flow and its numerical solution*, Numer. Methods Partial Differential Equations**5**(1989), no. 1, 1–24. MR**1012225**, 10.1002/num.1690050102**6.**S. H. Chou,*Analysis and convergence of a covolume method for the generalized Stokes problem*, Math. Comp.**66**(1997), no. 217, 85–104. MR**1372003**, 10.1090/S0025-5718-97-00792-8**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, 1850-1861, (1998). CMP**98:17****9.**S. H. Chou, D. Y. Kwak and P.S. Vassilevski,*Mixed upwinding covolume methods on rectangular grids for convection-diffusion 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**, 991-1011, (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**(1998), no. 2, 494–507 (electronic). MR**1618834**, 10.1137/S0036142996299964**12.**S. H. Chou and D. Y. Kwak,*Analysis and convergence of a MAC-like scheme for the generalized Stokes problem*, Numer. Methods Partial Differential Equations**13**(1997), no. 2, 147–162. MR**1436612**, 10.1002/(SICI)1098-2426(199703)13:2<147::AID-NUM2>3.0.CO;2-P**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**(1991), no. 2, 145–164. MR**1123813**, 10.1016/0045-7930(91)90017-C**14.**C. A. Hall, T. A. Porsching and P. Hu,*Covolume-dual variable method for thermally expandable flow on unstructured triangular grids*,**2**, Comp. Fluid Dyn, (1994), 111-139.**15.**Rong Hua Li,*Generalized difference methods for a nonlinear Dirichlet problem*, SIAM J. Numer. Anal.**24**(1987), no. 1, 77–88. MR**874736**, 10.1137/0724007**16.**R. H. Li and Z. Y. Chen,*The Generalized difference method for differential equations*, Jilin University Publishing House, (1994). (In Chinese)**17.**R. H. Li and P. Q. Zhu,*Generalized difference methods for second order elliptic partial differential equations (I)*, A Journal of Chinese Universities, (1982), 140-152.**18.**R. A. Nicolaides,*Direct discretization of planar div-curl problems*, SIAM J. Numer. Anal.**29**(1992), no. 1, 32–56. MR**1149083**, 10.1137/0729003**19.**R. A. Nicolaides,*Analysis and convergence of the MAC scheme. I. The linear problem*, SIAM J. Numer. Anal.**29**(1992), no. 6, 1579–1591. MR**1191137**, 10.1137/0729091**20.**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), 279-299.**21.**R. A. Nicolaides and X. Wu,*Analysis and convergence of the MAC scheme. II. Navier-Stokes equations*, Math. Comp.**65**(1996), no. 213, 29–44. MR**1320897**, 10.1090/S0025-5718-96-00665-5**22.**Endre Süli,*Convergence of finite volume schemes for Poisson’s equation on nonuniform meshes*, SIAM J. Numer. Anal.**28**(1991), no. 5, 1419–1430. MR**1119276**, 10.1137/0728073**23.**A. H. Schatz, V. C. Thomée, and L. B. Wahlbin,*Maximum norm stability and error estimates in parabolic finite element equations*, Comm. Pure Appl. Math.**33**(1980), no. 3, 265–304. MR**562737**, 10.1002/cpa.3160330305**24.**T. A. Porsching,*A network model for two-fluid flow*, Numer. Meth. Partial Diff. Eqns,**1**, (1985), 295-313.**25.**T. A. Porsching,*Error estimates for MAC-like approximations to the linear Navier-Stokes equations*, Numer. Math.**29**(1977/78), no. 3, 291–306. MR**0471622****26.**Ridgway Scott,*Optimal 𝐿^{∞} estimates for the finite element method on irregular meshes*, Math. Comp.**30**(1976), no. 136, 681–697. MR**0436617**, 10.1090/S0025-5718-1976-0436617-2**27.**Qiding Zhu,*A survey of superconvergence techniques in finite element methods*, Finite element methods (Jyväskylä, 1997) Lecture Notes in Pure and Appl. Math., vol. 196, Dekker, New York, 1998, pp. 287–302. MR**1602742**

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
65F10,
65N20,
65N30

Retrieve articles in all journals with MSC (1991): 65F10, 65N20, 65N30

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:
https://doi.org/10.1090/S0025-5718-99-01192-8

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