A general mixed covolume framework

for constructing conservative schemes

for elliptic problems

Authors:
So-Hsiang Chou and Panayot S. Vassilevski

Journal:
Math. Comp. **68** (1999), 991-1011

MSC (1991):
Primary 65F10, 65N20, 65N30

Published electronically:
February 23, 1999

MathSciNet review:
1648371

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 one-to-one transfer operator between trial and test spaces. In the nonsymmetric case (convection-diffusion equation) we show one-half order convergence rate for the flux variable which is approximated either by the lowest order Raviart-Thomas 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.

**1.**Douglas N. Arnold and Richard S. Falk,*A uniformly accurate finite element method for the Reissner-Mindlin plate*, SIAM J. Numer. Anal.**26**(1989), no. 6, 1276–1290. MR**1025088**, 10.1137/0726074**2.**Douglas N. Arnold, Richard S. Falk, and R. Winther,*Preconditioning in 𝐻(𝑑𝑖𝑣) and applications*, Math. Comp.**66**(1997), no. 219, 957–984. MR**1401938**, 10.1090/S0025-5718-97-00826-0**3.**Owe Axelsson,*Iterative solution methods*, Cambridge University Press, Cambridge, 1994. MR**1276069****4.**R. Beauwens and P. de Groen (eds.),*Iterative methods in linear algebra*, North-Holland Publishing Co., Amsterdam; International Association for Mathematics and Computers in Simulation (IMACS), New Brunswick, NJ, 1992. MR**1159713****5.**James H. Bramble and Joseph E. Pasciak,*A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems*, Math. Comp.**50**(1988), no. 181, 1–17. MR**917816**, 10.1090/S0025-5718-1988-0917816-8**6.**James H. Bramble, Joseph E. Pasciak, and Apostol T. Vassilev,*Analysis of the inexact Uzawa algorithm for saddle point problems*, SIAM J. Numer. Anal.**34**(1997), no. 3, 1072–1092. MR**1451114**, 10.1137/S0036142994273343**7.**Zhi Qiang Cai, Charles I. Goldstein, and Joseph E. Pasciak,*Multilevel iteration for mixed finite element systems with penalty*, SIAM J. Sci. Comput.**14**(1993), no. 5, 1072–1088. MR**1232176**, 10.1137/0914065**8.**Z. Cai, J. E. Jones, S. F. McCormick and T. F. Russell,*Control-Volume mixed finite element methods*, Computational Geosciences**1**(1997), 289-315.**9.**G. F. Carey, A. I. Pehlivanov, and P. S. Vassilevski,*Least-squares mixed finite element methods for non-selfadjoint elliptic problems. II. Performance of block-ILU factorization methods*, SIAM J. Sci. Comput.**16**(1995), no. 5, 1126–1136. MR**1346297**, 10.1137/0916065**10.**J. C. Cavendish, C. A. Hall, and T. A. Porsching,*A complementary volume approach for modelling three-dimensional Navier-Stokes equations using dual Delaunay/Voronoĭ tessellations*, Internat. J. Numer. Methods Heat Fluid Flow**4**(1994), no. 4, 329–345. MR**1286396**, 10.1108/EUM0000000004109**11.**Q. Du, R. Nicolaides, and X. Wu,*Analysis and convergence of a covolume approximation of the Ginzburg-Landau model of superconductivity*, SIAM J. Num. Anal.**35**(1998), 1049-1072. CMP**98:11****12.**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**13.**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**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), 497-507. 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), 1850-1861. CMP**98:17****19.**S. H. Chou and P. Vassilevski,*An upwinding cell-centered method with piecewise constant velocity over covolumes*, Numer. Meth. Partial Diff. Eqns. (1997), to appear.**20.**C. A. Hall and T. A. Porsching,*A characteristic-like method for thermally expandable flow on unstructured triangular grids*, Internat. J. Numer. Methods Fluids**22**(1996), no. 8, 731–754. MR**1387508**, 10.1002/(SICI)1097-0363(19960430)22:8<731::AID-FLD376>3.3.CO;2-9**21.**C. A. Hall, T. A. Porsching and P. Hu,*Covolume-dual 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 convection-diffusion 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), 279-299.**25.**Roy A. Nicolaides and Xiaonan Wu,*Covolume solutions of three-dimensional div-curl equations*, SIAM J. Numer. Anal.**34**(1997), no. 6, 2195–2203. MR**1480375**, 10.1137/S0036142994277286**26.**Torgeir Rusten and Ragnar Winther,*A preconditioned iterative method for saddlepoint problems*, SIAM J. Matrix Anal. Appl.**13**(1992), no. 3, 887–904. Iterative methods in numerical linear algebra (Copper Mountain, CO, 1990). MR**1168084**, 10.1137/0613054**27.**Y. Saad,*Iterative Methods for Sparse Linear Systems*, PSW Kent, 1995.**28.**David Silvester and Andrew Wathen,*Fast iterative solution of stabilised Stokes systems. II. Using general block preconditioners*, SIAM J. Numer. Anal.**31**(1994), no. 5, 1352–1367. MR**1293519**, 10.1137/0731070**29.**Panayot S. Vassilevski and Raytcho D. Lazarov,*Preconditioning mixed finite element saddle-point elliptic problems*, Numer. Linear Algebra Appl.**3**(1996), no. 1, 1–20. MR**1373366**, 10.1002/(SICI)1099-1506(199601/02)3:1<1::AID-NLA67>3.3.CO;2-5**30.**Panayot S. Vassilevski and Jun Ping Wang,*Multilevel iterative methods for mixed finite element discretizations of elliptic problems*, Numer. Math.**63**(1992), no. 4, 503–520. MR**1189534**, 10.1007/BF01385872**31.**Panayot S. Vassilevski,*On two ways of stabilizing the hierarchical basis multilevel methods*, SIAM Rev.**39**(1997), no. 1, 18–53. MR**1439484**, 10.1137/S0036144595282211

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, 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:
https://doi.org/10.1090/S0025-5718-99-01090-X

Keywords:
Conservative schemes,
mixed finite elements,
covolume methods,
finite volume methods,
finite volume element,
Raviart--Thomas 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