An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems

International Journal for Numerical Methods in Engineering

Volumn 76, Issue 11, 2008, Pages 1696-1723

  Beirão da Veiga, Lourenço ^a   Manzini, Gianmarco ^b  

^a UNIVERSITY OF MILAN   (Italy)

^b CNR   (Italy)

Author keywords

A posteriori estimates; Adaptivity; Elliptic problems; Mimetic finite differences

Indexed keywords

FINITE DIFFERENCE METHOD;

A POSTERIORI ESTIMATES; ADAPTIVITY; COMPUTATIONAL DOMAINS; COMPUTATIONAL MESHES; DIFFUSION EQUATIONS; ELLIPTIC PROBLEMS; FINITE DIFFERENCE APPROXIMATIONS; MIMETIC FINITE DIFFERENCES; PERMEABILITY TENSORS; POSTERIORI ERROR ESTIMATORS; POSTERIORI ERROR INDICATORS; TYPE ERRORS;

COMPUTATIONAL MECHANICS;

EID: 60949113471     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2377     Document Type: Article

References (30)

1. Hyman JM, Shashkov M. Adjoint operators for the natural discretizations of the divergence, gradient and curl on logically rectangular grids. Applied Numerical Mathematics 1997; 25(4):413-442.
2. Hyman JM, Shashkov M. Approximation of boundary conditions for mimetic finite-difference methods. Computers and Mathematics with Applications 1998; 36(5):79-99.
3. Hyman JM, Shashkov M. The orthogonal decomposition theorems for mimetic finite difference methods. SIAM Journal on Numerical Analysis 1999; 36(3):788-818.
4. Shashkov M, Steinberg S. Solving diffusion equations with rough coefficients in rough grids. Journal of Computational Physics 1996; 129(2):383-405.
5. Domelevo K, Omnes P. A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM: Mathematical Modelling and Numerical Analysis (M2AN) 2005; 39(6):1203-1249.
6. Kuznetsov Y, Lipnikov K, Shashkov M. The mimetic finite difference method on polygonal meshes for diffusion-type problems. Computational Geosciences 2005; 8:301-324.
7. Lipnikov K, Morel J, Shashkov M. Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes. Journal of Computational Physics 2004; 199:589-597.
8. Lipnikov K, Shashkov M, Svyatskiy D. The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes. Journal of Computational Physics 2006; 211:473-491.
9. Hyman J, Shashkov M. Mimetic discretisations of Maxwell's equations and the equations of magnetic diffusion. Progress in Electromagnetic Research 2001; 32:89-121.
10. Hyman J, Morel J, Shashkov M, Steinberg S. Mimetic finite difference methods for diffusion equations. Computational Geosciences 2002; 6(3-4):333-352.
11. Campbell JC, Shashkov MJ. A tensor artificial viscosity using a mimetic finite difference algorithm. Journal of Computational Physics 2001; 172(2):739-765.
12. Berndt M, Lipnikov K, Moulton JD, Shashkov M. Convergence of mimetic finite difference discretizations of the diffusion equation. Numerische Mathematik 2001; 9:253-284.
13. Berndt M, Lipnikov K, Shashkov M, Wheeler MF, Yotov I. Superconvergence of the velocity in mimetic finite difference methods on quadrilaterals. SIAM Journal on Numerical Analysis 2005; 43:1728-1749.
14. Brezzi F, Lipnikov K, Shashkov M. Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM Journal on Numerical Analysis 2005; 43:1872-1896.
15. Brezzi F, Lipnikov K, Shashkov M. Convergence of mimetic finite difference methods for diffusion problems on polyhedral meshes with curved faces. Mathematical Models and Methods in Applied Sciences 2006; 275-298.
16. Brezzi F, Lipnikov K, Shashkov M, Simoncini V. A new discretization methodology for diffusion problems on generalized polyhedral meshes. Computer Methods in Applied Mechanics and Engineering 2007; 196: 3682-3692.
17. Brezzi F, Lipnikov K, Simoncini V. A family of mimetic finite difference methods on polygonal and polyhedral meshes. Mathematical Models and Methods in Applied Sciences 2005; 15:1533-1553.
18. Beirão da Veiga L. A residual based error estimator for the mimetic finite difference method. Numerische Mathematik 2008; 108(3):387-406.
19. Agmon S. Lectures on Elliptic Boundary Value Problems. Van Nostrand: Princeton, NJ, 1965.
20. Beirão da Veiga L, Manzini G. An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems with general diffusion tensors. Technical Report 17PV07/17/0, IMATI-CNR, Pavia, Italy, 2007.
21. Cangiani A, Manzini G. Flux reconstruction and pressure post-processing in mimetic finite difference methods. Computer Methods in Applied Mechanics and Engineering 2008; 197(9-12):933-945.
22. Ainsworth M, Oden JT. A posteriori error estimation in finite element analysis. Computer Methods in Applied Mechanics and Engineering 1997; 142(1-2):1-88.
23. Ainsworth M, Oden JT. Error Estimation in Finite Element Analysis. Wiley: New York, 2000.
24. Verfürth R. A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement. Wiley and Teubner: Stuttgart, 1996.
25. Brenner SC, Scott LR. The Mathematical Theory of Finite Element Methods. Springer: Berlin, 1994.
26. Ciarlet PG. The Finite Element Method for Elliptic Problems. North-Holland: Amsterdam, 1978.
27. Carstensen C. A posteriori error estimate for the mixed finite element method. Mathematics of Computation 1996; 66:465-476.
28. Lovadina C, Stenberg R. Energy norm a posteriori error estimates for mixed finite element methods. Mathematics of Computation 2006; 75:1659-1674.
29. Riviere B, Wheeler MF, Banas K. Discontinuous Galerkin method applied to a single phase flow in porous media. Computational Geosciences 2000; 4:337-349.
30. Vohralik M. A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations. SIAM Journal on Numerical Analysis 2007; 45:1570-1599.