A residual based error estimator for the Mimetic Finite Difference method

Numerische Mathematik

Volumn 108, Issue 3, 2008, Pages 387-406

  Beirao Da Veiga L ^a  

^a Dipartimento di Matematica F Enriques   (Italy)

Author keywords

65N30

Indexed keywords

EID: 37449025866     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-007-0126-6     Document Type: Article

References (26)

1. Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, London (2000)
2. Agmon, S.: Lectures on Elliptic Boundary Value Problems. Van Nostrand, Princeton (1965)
3. Arnold, D.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742-760 (1982)
4. Berndt, M., Lipnikov, K., Moulton, J.D., Shashkov, M.: Convergence of mimetic finite difference discretizations of the diffusion equation. J. Numer. Math. 9, 253-284 (2001)
5. 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)
6. Braess, D., Verfürth, R.: A posteriori error estimators for the Raviart-Thomas element. SIAM J. Numer. Anal. 33, 2431-2444 (1996)
7. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, Heidelberg (1994)
8. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)
9. 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)
10. Brezzi, F., Lipnikov, K., Simoncini, V.: A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15, 1533-1553 (2005)
11. Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of mimetic finite difference methods for diffusion problems on polyhedral meshes with curved faces. Math. Models Methods Appl. Sci. 16, 275-298 (2006)
12. Brezzi, F., Lipnikov, K., Shashkov, M., Simoncini, V.: A new discretization methodology for diffusion problems on generalized polyhedral meshes. Comput. Meth. Appl. Mech. Eng. 196(37-40), 3682-3692 (2007)
13. Cangiani, A., Manzini, G.: Post-processing of pressure and flux in mimetic finite differences. Preprint IMATI-CNR, 48-PV (2007)
14. Carstensen, C.: A posteriori error estimate for the mixed finite element method. Math. Comput. 66, 465-476 (1996)
15. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
16. Girault, G., Raviart, P.: Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer, Heidelberg (1986)
17. Hyman, J., Shashkov, M., Steinberg, M.: The numerical solution of diffusion problems in strongly heterogeneus non-isotropic materials. J. Comput. Phys. 132, 130-148 (1997)
18. Kuznetsov, Y., Lipnikov, K., Shashkov, M.: The mimetic finite difference method on polygonal meshes for diffusion-type problems. Comput. Geosci. 8, 301-324 (2005)
19. Lipnikov, K., Morel, J., Shashkov, M.: Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes. J. Comput. Phys. 199, 589-597 (2004)
20. Lipnikov, K., Shashkov, M., Svyatskiy, D.: The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes. J. Comput. Phys. 211, 473-491 (2006)
21. Liska, R., Shashkov, M., Ganza, V.: Analysis and optimization of inner products for mimetic finite difference methods on triangular grid. Math. Comput. Simul. 67, 55-66 (2004)
22. Lovadina, C., Stenberg, R.: Energy norm a posteriori error estimates for mixed finite element methods. Math. Comput. 75, 1659-1674 (2006)
23. Morel, J., Hall, M., Shaskov, M.: A local support-operators diffusion discretization scheme for hexahedral meshes. J. Comput. Phys. 170, 338-372 (2001)
24. 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)
25. Stenberg, R.: Postprocessing schemes for some mixed finite elements. Math. Model. Numer. Anal. 25, 151-168 (1991)
26. Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh Refinement. Wiley and Teubner, Stuttgart (1996)