A general mixed covolume framework for constructing conservative schemes for elliptic problems
Authors:
SoHsiang Chou and Panayot S. Vassilevski
Journal:
Math. Comp. 68 (1999), 9911011
MSC (1991):
Primary 65F10, 65N20, 65N30
Published electronically:
February 23, 1999
MathSciNet review:
1648371
Fulltext PDF Free Access
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Abstract: We present a general framework for the finite volume or covolume schemes developed for second order elliptic problems in mixed form, i.e., written as first order systems. We connect these schemes to standard mixed finite element methods via a onetoone transfer operator between trial and test spaces. In the nonsymmetric case (convectiondiffusion equation) we show onehalf order convergence rate for the flux variable which is approximated either by the lowest order RaviartThomas space or by its image in the space of discontinuous piecewise constants. In the symmetric case (diffusion equation) a first order convergence rate is obtained for both the state variable (e.g., concentration) and its flux. Numerical experiments are included.
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 2.
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 5.
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 6.
 J. H. Bramble, J. E. Pasciak, and A. T. Vassilev, Analysis of inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal. 34 (1997), 10721092. MR 98c:65182
 7.
 Z. Cai, C. I. Goldstein and J. E. Pasciak, Multilevel iteration for mixed finite element systems with penalty, SIAM J. Sci. Comput. 14 (1993), 10721088. MR 94h:65116
 8.
 Z. Cai, J. E. Jones, S. F. McCormick and T. F. Russell, ControlVolume mixed finite element methods, Computational Geosciences 1 (1997), 289315.
 9.
 G. F. Carey, A. I. Pehlivanov, and P. S. Vassilevski, Leastsquares mixed finite elements for nonselfadjoint elliptic problems: II, Performance of blockILU factorization methods, SIAM J. Sci. Comput. 16 (1995), 11261136. MR 97f:65069
 10.
 J. C. Cavendish, C. A. Hall and T. A. Porsching, A complementary volume approach for modeling threedimensional NavierStokes equations using dual Delaunay/Voronoi tessellations, Internat. J. Numer. Methods Heat Fluid Flow, 4 (1994) 329345. MR 95d:76088
 11.
 Q. Du, R. Nicolaides, and X. Wu, Analysis and convergence of a covolume approximation of the GinzburgLandau model of superconductivity, SIAM J. Num. Anal. 35 (1998), 10491072. CMP 98:11
 12.
 S. H. Chou, Analysis and convergence of a covolume method for the generalized Stokes problem, Math. Comp. 66 (1997), 85104. MR 97e:65109
 13.
 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
 14.
 S. H. Chou and D. Y. Kwak, Mixed covolume methods on rectangular grids for elliptic problems, SIAM J. Num. Anal. (1998), to appear.
 15.
 S. H. Chou and Q. Li, Error estimates in and in covolume methods for elliptic and parabolic problems: A unified approach, Math. Comp. (1996), submitted.
 16.
 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), 497507. CMP 98:11
 17.
 S. H. Chou, D. Y. Kwak and P. Vassilevski, Mixed covolume methods on rectangular grids for convection dominated problems, SIAM J. Sci. Computing, (1998), to appear.
 18.
 S. H. Chou, D. Y. Kwak and P. Vassilevski, Mixed covolume methods for elliptic problems on triangular grids, SIAM J. Numer. Anal. 35 (1998), 18501861. CMP 98:17
 19.
 S. H. Chou and P. Vassilevski, An upwinding cellcentered method with piecewise constant velocity over covolumes, Numer. Meth. Partial Diff. Eqns. (1997), to appear.
 20.
 C. A. Hall and T. A. Porsching, A characteristiclike method for thermally expandable flow on unstructured triangular grids, Internat. J. Numer. Methods Fluids 22 (1996), 731754. MR 97d:76029
 21.
 C. A. Hall, T. A. Porsching and P. Hu, Covolumedual variable method for thermally expandable flow on unstructured triangular grids, Comp. Fluid Dyn. 2 (1994).
 22.
 F. H. Harlow and F. E. Welch, Numerical calculations of time dependent viscous incompressible flow of fluid with a free surface, Phys. Fluids 8 (1965), 2181.
 23.
 M. Liu, J. Wang and N. Yan, New error estimates for approximate solutions of convectiondiffusion problems by mixed and discontinuous Galerkin methods, (1997) preprint.
 24.
 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.
 25.
 R. Nicolaide and X. Wu,Covolume solutions of threedimensional divcurl equations, SIAM. J. Numer. Anal. 34 (1997), 21952203. MR 98f:65096
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 T. Rusten and R. Winther, A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl. 13 (1992), 887904. MR 93a:65043
 27.
 Y. Saad, Iterative Methods for Sparse Linear Systems, PSW Kent, 1995.
 28.
 D. Silvester and A. Wathen, Fast iterative solution of stabilised Stokes systems, II. Using general block preconditioners. SIAM J. Numer. Anal. 31 (1994), 13521367. MR 95g:65132
 29.
 P. S. Vassilevski and R. D. Lazarov, Preconditioning mixed finite element saddlepoint elliptic problems, Numer. Linear Alg. Appl. 3 (1996), 120. MR 96m:65111
 30.
 P. S. Vassilevski and J. Wang, Multilevel iterative methods for mixed finite element discretizations of elliptic problems, Numer. Math. 63 (1992), 503520. MR 93j:65187
 31.
 P. S. Vassilevski, On two ways of stabilizing the hierarchical basis methods, SIAM Rev. 39 (1997), 1853. MR 98a:65178
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Additional Information
SoHsiang Chou
Affiliation:
Department of Mathematics, Bowling Green State University, Bowling Green, Ohio 43403, U.S.A.
Email:
chou@zeus.bgsu.edu
Panayot S. Vassilevski
Affiliation:
Center of Informatics and Computing Technology, Bulgarian Academy of Sciences, “Acad. G. Bontchev” street, Block 25 A, 1113 Sofia, Bulgaria
Email:
panayot@iscbg.acad.bg
DOI:
http://dx.doi.org/10.1090/S002557189901090X
PII:
S 00255718(99)01090X
Keywords:
Conservative schemes,
mixed finite elements,
covolume methods,
finite volume methods,
finite volume element,
RaviartThomas spaces,
error estimates,
$H(\mydiv)$preconditioning
Received by editor(s):
June 16, 1997
Published electronically:
February 23, 1999
Additional Notes:
The work of the second author was partially supported by the Bulgarian Ministry for Education, Science and Technology under grant I–95 # 504
Article copyright:
© Copyright 1999
American Mathematical Society
