Mimetic finite difference methods for diffusion equations

Computational Geosciences

Volumn 6, Issue 3-4, 2002, Pages 333-352

  Hyman, J ^a   Morel, J ^a   Shashkov, M ^a   Steinberg, S ^b  

^a LOS ALAMOS NATIONAL LABORATORY   (United States)

^b UNIVERSITY OF NEW MEXICO   (United States)

Author keywords

Diffusion equations; Locally conservative method; Mimetic discretizations

Indexed keywords

DIFFUSION; FINITE DIFFERENCE TECHNIQUE; FLUID FLOW; MATHEMATICAL METHOD; POROUS MEDIUM;

ANISOTROPIC MEDIUM; DIFFUSION; FINITE DIFFERENCE METHOD; FLUID FLOW; HETEROGENEOUS MEDIUM; NUMERICAL METHOD; POROUS MEDIUM;

ANISOTROPY; DIFFUSION; FINITE ELEMENT ANALYSIS; MATERIALS; MATHEMATICAL ANALYSIS;

EID: 0036923031     PISSN: 14200597     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1021282912658     Document Type: Article

References (24)

1. T. Arbogast, C.N. Dawson, P.T. Keenan, M.F. Wheeler and I. Yotov, Enhanced cell-centered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comput. 18 (1997) 1-32.
2. D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: Implementation, post-processing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985) 7-32.
3. Z. Cai, J.E. Jones, S.F. McCormick and T.F. Russel, Control-volume mixed finite element methods, Comput. Geosci. 1 (1997) 289-315.
4. E.J. Caramana, D.E. Burton, M.J. Shashkov and P.P. Whalen, The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, J. Comput. Phys. 146 (1998) 227-262.
5. J.E. Dendy, Jr., Black box multigrid, J. Comput. Phys. 46 (1982) 366-386.
6. J.E. Dendy and J.D. Moulton, Some aspects of multigrid for mixed discretizations, in: Lecture Notes in Computational Science and Engineering, Vol. 14, eds. E. Dick, K. Riemslagh and J. Vierendeels (Springer, New York, 1999) pp. 80-86.
7. R. Hiptmair and R.H.W. Hoppe, Multilevel methods for mixed finite elements in three dimensions, Numer. Math. 82 (1999) 253-279.
8. J. Hyman and M. Shashkov, Mimetic discretizations for Maxwell's equations and the equations of magnetic diffusion, PIER 32 (2001) 89-121.
9. J. Hyman, M. Shashkov and S. Steinberg, The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials, Los Alamos National Laboratory Report LA-UR-96-532 (1996).
10. J. Hyman, M. Shashkov and S. Steinberg, The effect of inner products for discrete vector fields on the accuracy of mimetic finite difference methods, Comput. Math. Appl. 42 (2001) 1527-1547.
11. J.M. Hyman and M. Shashkov, Mimetic discretizations for Maxwell's equations, J. Comput. Phys. 151 (1999) 881-909.
12. J.M. Hyman and M. Shashkov, Natural discretizations for the divergence, gradient, and curl on logically rectangular grids, Comput. Math. Appl. 33 (1997) 81-104.
13. J.M. Hyman and M. Shashkov, The adjoint operators for the natural discretizations for the divergence, gradient, and curl on logically rectangular grids. Appl. Numer. Math. 25 (1997) 413-442.
14. J.M. Hyman and M. Shashkov, The approximation of boundary conditions for mimetic finite difference methods, Comput. Math. Appl. 36 (1998) 79-99.
15. J.M. Hyman and M. Shashkov, The orthogonal decomposition theorems for mimetic finite difference methods, SIAM J. Numer. Anal. 36 (1999) 788-818.
16. J.M. Hyman, M. Shashkov and S. Steinberg, The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials, J. Comput. Phys. 132 (1997) 130-148.
17. L.G. Margolin and J.J. Pyun, A method for treating hourglass pattern, LA-UR-87-439, Report of Los Alamos National Laboratory, Los Alamos, NM, USA.
18. L. Margolin, M. Shashkov and P. Smolarkiewicz, A discrete operator calculus for finite difference approximations, Comput. Methods Appl. Mech. Engrg. 187 (2000) 365-383.
19. J. Morel, M. Hall and M. Shashkov, A local support-operators diffusion discretization scheme for hexahedral meshes, J. Comput. Phys. 170 (2001) 338-372.
20. J. Morel, R. Roberts and M. Shashkov, A local support-operators diffusion discretization scheme for quadrilateral r-z meshes, J. Comput. Phys. 144 (1998) 17-51.
21. M. Shashkov. Conservative Finite-Difference Schemes on General Grids (CRC Press, Boca Raton, FL, 1995).
22. M. Shashkov and S. Steinberg, Support-operator finite-difference algorithms for general elliptic problems, J. Comput. Phys. 118 (1995) 131-151.
23. M. Shashkov and S. Steinberg, Solving diffusion equations with rough coefficients in rough grids, J. Comput. Phys. 129 (1996) 383-405.
24. U. Trottenberg, C.W. Oosterlee and A. Schüller, Multigrid (Academic Press, San Diego, CA, 2001).