A general mixed covolume framework for constructing conservative schemes for elliptic problems

Mathematics of Computation

Volumn 68, Issue 227, 1999, Pages 991-1011

  Chou, So Hsiang ^a   Vassilevski, Panayot S ^b  

^a BOWLING GREEN STATE UNIVERSITY   (United States)

^b BULGARIAN ACADEMY OF SCIENCES   (Bulgaria)

Author keywords

Conservative schemes; Covolume methods; Error estimates; Finite volume element; Finite volume methods; H(div) preconditioning; Mixed finite elements; Raviart Thomas spaces

Indexed keywords

EID: 0039243425     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0025-5718-99-01090-x     Document Type: Article

References (52)

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 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
3. D. N. Arnold, R. S. Falk and R. Winther, Preconditioning in H(div) and applications, Math. Comp. 66 (1997), 957-984. MR 971:65177
4. D. N. Arnold, R. S. Falk and R. Winther, Preconditioning in H(div) and applications, Math. Comp. 66 (1997), 957-984. MR 971:65177
5. O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, 1994. MR 95f:65005
6. O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, 1994. MR 95f:65005
7. 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
8. 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
9. 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
10. 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
11. 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
12. 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
13. 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
14. 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
15. Z. Cai, J. E. Jones, S. F. McCormick and T. F. Russell, Control-Volume mixed finite element methods, Computational Geosciences 1 (1997), 289-315.
16. G. F. Carey, A. I. Pehlivanov, and P. S. Vassilevski, Least-squares mixed finite elements for nonselfadjoint elliptic problems: II, Performance of block-IL U factorization methods, SIAM J. Sci. Comput. 16 (1995), 1126-1136. MR 97f:65069
17. G. F. Carey, A. I. Pehlivanov, and P. S. Vassilevski, Least-squares mixed finite elements for nonselfadjoint elliptic problems: II, Performance of block-IL U factorization methods, SIAM J. Sci. Comput. 16 (1995), 1126-1136. MR 97f:65069
18. 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
19. 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
20. 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
21. 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
22. S. H. Chou, Analysis and convergence of a covolume method for the generalized Stokes problem, Math. Comp. 66 (1997), 85-104. MR 97e:65109
23. S. H. Chou, Analysis and convergence of a covolume method for the generalized Stokes problem, Math. Comp. 66 (1997), 85-104. MR 97e:65109
24. 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
25. 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
26. S. H. Chou and D. Y. Kwak, Mixed covolume methods on rectangular grids for elliptic problems, SIAM J. Num. Anal. (1998), to appear.
27. ∞ in covolume methods for elliptic and parabolic problems: A unified approach, Math. Comp. (1996), submitted.
28. 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
29. 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
30. 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.
31. 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
32. 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
33. 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.
34. 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
35. 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
36. 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).
37. 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.
38. M. Liu, J. Wang and N. Yan, New error estimates for approximate solutions of convectiondiffusion problems by mixed and discontinuous Galerkin methods, (1997) preprint.
39. 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.
40. R. Nicolaide and X. Wu, Covolume solutions of three-dimensional div-curl equations, SIAM. J. Numer. Anal. 34 (1997), 2195-2203. MR 98f:65096
41. R. Nicolaide and X. Wu, Covolume solutions of three-dimensional div-curl equations, SIAM. J. Numer. Anal. 34 (1997), 2195-2203. MR 98f:65096
42. T. Rusten and R. Winther, A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl. 13 (1992), 887-904. MR 93a:65043
43. T. Rusten and R. Winther, A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl. 13 (1992), 887-904. MR 93a:65043
44. Y. Saad, Iterative Methods for Sparse Linear Systems, PSW Kent, 1995.
45. 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
46. 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
47. 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
48. 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
49. 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
50. 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
51. P. S. Vassilevski, On two ways of stabilizing the hierarchical basis methods, SIAM Rev. 39 (1997), 18-53. MR 98a:65178
52. P. S. Vassilevski, On two ways of stabilizing the hierarchical basis methods, SIAM Rev. 39 (1997), 18-53. MR 98a:65178