Local flux mimetic finite difference methods

Numerische Mathematik

Volumn 112, Issue 1, 2009, Pages 115-152

  Lipnikov, Konstantin ^a   Shashkov, Mikhail ^a   Yotov, Ivan ^b  

^a LOS ALAMOS NATIONAL LABORATORY   (United States)

^b UNIVERSITY OF PITTSBURGH   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77949332778     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-008-0203-5     Document Type: Article

References (47)

1. Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comp. Geosci. 6, 405-432 (2002).
2. Aavatsmark, I., Barkve, T., Bøe, O., Mannseth, T.: Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods. SIAM J. Sci. Comput. 19, 1700-1716 (1998).
3. Adams, R. A.: Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York, London. Pure Appl. Math., vol. 65 (1975).
4. Arbogast, T., Dawson, C. N., Keenan, P. T., Wheeler, M. F., Yotov, I.: Enhanced cell-centered finite differences for elliptic equations on general geometry. SIAM J. Sci. Comput. 19, 404-425 (1998).
5. Arbogast, T., Wheeler, M. F., Yotov, I.: Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 34, 828-852 (1997).
6. Arnold, D. N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742-760 (1982).
7. Arnold, D. N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modèl. Math. Anal. Numèr. 19, 7-32 (1985).
8. Baranger, J., Maitre, J.-F., Oudin, F.: Connection between finite volume and mixed finite element methods. RAIRO Modél. Math. Anal. Numér. 30, 445-465 (1996).
9. Berndt, M., Lipnikov, K., Moulton, J. D., Shashkov, M.: Convergence of mimetic finite difference discretizations of the diffusion equation. East-West J. Numer. Math. 9, 253-284 (2001).
10. Berndt, M., Lipnikov, K., Shashkov, M., Wheeler, M. F., Yotov, I.: A mortar mimetic finite difference method on non-matching grids. Numer. Math. 102, 203-230 (2005).
11. Berndt, M., Lipnikov, K., Shashkov, M., Wheeler, M. F., Yotov, I.: Superconvergence of the velocity in mimetic finite difference methods on quadrilaterals. SIAM J. Numer. Anal. 43, 1728-1749 (2005).
12. Breil, J., Maire, P.-H.: A cell-centered diffusion scheme on two-dimensional unstructured meshes. J. Comput. Phys. 224, 785-823 (2007).
13. Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods. Springer, Berlin (1994).
14. Brezzi, F., Douglas, J. Jr., Marini, L. D.: Two families of mixed elements for second order elliptic problems. Numer. Math. 88, 217-235 (1985).
15. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991).
16. Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43, 1872-1896 (2005).
17. Brezzi, F., Lipnikov, K., Simoncini, V.: A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Mod. Methods. Appl. Sci. 15, 1533-1552 (2005).
18. Campbell, J., Shashkov, M.: A tensor artificial viscosity using a mimetic finite difference algorithm. J. Comput. Phys. 172, 739-765 (2001).
19. Ciarlet, P. G.: The Finite Element Method for Elliptic Problems. North-Holland, New York (1978).
20. Crumpton, P., Shaw, G., Ware, A.: Discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients. J. Comput. Phys. 116, 343-358 (1995).
21. Edwards, M. G., Rogers, C. F.: Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geosci. 2, 259-290 (1998).
22. Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of numerical analysis, vol. VII. Handb. Numer. Anal., VII, pp. 713-1020. North-Holland, Amsterdam (2000).
23. Eymard, R., Gallouët, T., Herbin, R.: A cell-centered finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension. IMA J. Numer. Anal. 26, 326-353 (2006).
24. Eymard, R., Gallouët, T., Herbin, R.: A new finite volume scheme for anisotropic diffusion problems on general grids: convergence analysis. C. R. Math. Acad. Sci. Paris 344, 403-406 (2007).
25. Grisvard, P.: Elliptic Problems in Nonsmooth domains. Pitman, London (1985).
26. Hyman, J., Shashkov, M.: The approximation of boundary conditions for mimetic finite difference methods. Comput. Math. Appl. 36, 79-99 (1998).
27. Hyman, J., Shashkov, M.: Mimetic discretizations for Maxwell's equations and the equations of magnetic diffusion. Prog. Electromagnetic Res. 32, 89-121 (2001).
28. Hyman, J., Shashkov, M., Steinberg, S.: The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials. J. Comput. Phys. 132, 130-148 (1997).
29. Klausen, R. A., Russell, T. F.: Relationships among some locally conservative discretization methods which handle discontinuous coefficients. Comput. Geosci. 8, 341-377 (2004).
30. Klausen, R. A., Winther, R.: Convergence of multipoint flux approximations on quadrilateral grids. Numer. Methods Partial Differ. Equations 22, 1438-1454 (2006).
31. Klausen, R. A., Winther, R.: Robust convergence of multi point flux approximation on rough grids. Numer. Math. 104, 317-337 (2006).
32. Kuznetsov, Y., Repin, S.: New mixed finite element method on polygonal and polyhedral meshes. Russian J. Numer. Anal. Math. Modelling 18, 261-278 (2003).
33. Kuznetsov, Y., Repin, S.: Convergence analysis and error estimates for mixed finite element method on distorted meshes. J. Numer. Math. 13, 33-51 (2005).
34. Le Potier, C.: Schéma volumes finis pour des opérateurs de diffusion fortement anisotropes sur des maillages non structurés. C. R. Math. Acad. Sci. Paris 340, 921-926 (2005).
35. Lions, J. L., Magenes, E.: Non-homogeneous boundary value problems and applications, vol. 1. Springer, Berlin (1972).
36. Lipnikov, K., Shashkov, M., Svyatskiy, D.: The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes. J. Comput. Phys. 211, 473-491 (2005).
37. Margolin, L., Shashkov, M., Smolarkiewicz, P.: Adiscrete operator calculus for finite difference approximations. Comput. Methods Appl. Mech. Eng. 187, 365-383 (2000).
38. Mathew, T.: Domain decomposition and iterative refinement methods for mixed finite element discretizations of elliptic problems. PhD thesis, Courant Institute of Mathematical Sciences, New York University, Tech. Rep. 463 (1989).
39. Micheletti, S., Sacco, R., Saleri, F.: On some mixed finite element methods with numerical integration. SIAM J. Sci. Comput. 23, 245-270 (2001).
40. Morel, J., Moulton, J., Shashkov, M.: Mimetic preconditioners for mixed discretizations of the diffusion equation. Tech. Report LA-UR-01-807, Los Alamos National Laboratory (2001). http://www. ima. umn. edu/talks/workshops/5-11-15. 2004/moulton/moulton. pdf.
41. Morel, J., Roberts, R., Shashkov, M.: A local support-operators diffusion discretization scheme for quadrilateral r-z meshes. J. Comput. Phys. 144, 17-51 (1998).
42. Raviart, P., Thomas, J.-M.: A mixed finite element method for second order elliptic problems. In: Galligani, I., Magenes, E., (eds.) Mathematical Aspects of the Finite Element Method, pp. 292-315. Springer, Berlin (1977).
43. Russell, T. F., Wheeler, M. F.: Finite element and finite difference methods for continuous flows in porous media. In: Ewing, R. E., (ed.) The Mathematics of Reservoir Simulation, vol. 1, pp. 35-106. Frontiers in Applied Mathematics. SIAM, Philadelphia, PA (1983).
44. Stüben, K.: Algebraic multigrid (AMG): experiences and comparisons. Appl. Math. Comput. 13, 419-452 (1983).
45. Temam, R.: Navier-Stokes equations. Theory and numerical analysis, Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, RI (2001).
46. Vassilevski, P. S., Petrova, S. I., Lazarov, R. D.: Finite difference schemes on triangular cell-centered grids with local refinement. SIAM J. Sci. Statist. Comput. 13, 1287-1313 (1992).
47. Wheeler, M. F., Yotov, I.: A multipoint flux mixed finite element method. SIAM J. Numer. Anal. 44, 2082-2106 (2006).