Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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


References [Enhancements On Off] (What's this?)

  • 1. D. N. Arnold and R. S. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal. 26 (1989), 1276-1290. MR 91c:65068
  • 2. D. N. Arnold, R. S. Falk and R. Winther, Preconditioning in H(div) and applications, Math. Comp. 66 (1997), 957-984. MR 97i:65177
  • 3. O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, 1994. MR 95f:65005
  • 4. O. Axelsson and P. S. Vassilevski, ``Construction of variable-step preconditioners for inner-outer iteration methods'', Proceedings of IMACS Conference on Iterative methods, April 1991, Brussels, Belgium, Iterative Methods in Linear Algebra, (R. Beauwens and P. de Groen, eds.), North Holland, 1992, 1-14. MR 92m:65053
  • 5. J. H. Bramble and J. E. Pasciak, A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems, Math. Comp. 50 (1988), 1-17. MR 89m:65097a
  • 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), 1072-1092. 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), 1072-1088. MR 94h:65116
  • 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 elements for nonselfadjoint elliptic problems: II, Performance of block-ILU factorization methods, SIAM J. Sci. Comput. 16 (1995), 1126-1136. MR 97f:65069
  • 10. J. C. Cavendish, C. A. Hall and T. A. Porsching, A complementary volume approach for modeling three-dimensional Navier-Stokes equations using dual Delaunay/Voronoi tessellations, Internat. J. Numer. Methods Heat Fluid Flow, 4 (1994) 329-345. MR 95d:76088
  • 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), 85-104. MR 97e:65109
  • 13. S. H. Chou and D. Y. Kwak, Analysis and convergence of a MAC-like scheme for the generalized Stokes problem, Numer. Meth. Partial Diff. Eqns. 13 (1997), 147-162. 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 $L^2, H^1$ and $L^\infty$ 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), 731-754. MR 97d:76029
  • 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. R. Nicolaide and X. Wu,Covolume solutions of three-dimensional div-curl equations, SIAM. J. Numer. Anal. 34 (1997), 2195-2203. MR 98f:65096
  • 26. T. Rusten and R. Winther, A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl. 13 (1992), 887-904. 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), 1352-1367. MR 95g:65132
  • 29. P. S. Vassilevski and R. D. Lazarov, Preconditioning mixed finite element saddle-point elliptic problems, Numer. Linear Alg. Appl. 3 (1996), 1-20. 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), 503-520. MR 93j:65187
  • 31. P. S. Vassilevski, On two ways of stabilizing the hierarchical basis methods, SIAM Rev. 39 (1997), 18-53. MR 98a:65178

Similar Articles

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

American Mathematical Society