Mimetic finite differences for elliptic problems
ESAIM: Modélisation mathématique et analyse numérique, Volume 43 (2009) no. 2, pp. 277-295.

We developed a mimetic finite difference method for solving elliptic equations with tensor coefficients on polyhedral meshes. The first-order convergence estimates in a mesh-dependent H 1 norm are derived.

DOI: 10.1051/m2an:2008046
Classification: 65N06, 65N12, 65N15, 65N30
Keywords: finite differences, polyhedral meshes, diffusion equation, error estimates
@article{M2AN_2009__43_2_277_0,
     author = {Brezzi, Franco and Buffa, Annalisa and Lipnikov, Konstantin},
     title = {Mimetic finite differences for elliptic problems},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {277--295},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {2},
     year = {2009},
     doi = {10.1051/m2an:2008046},
     mrnumber = {2512497},
     zbl = {1177.65164},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2008046/}
}
TY  - JOUR
AU  - Brezzi, Franco
AU  - Buffa, Annalisa
AU  - Lipnikov, Konstantin
TI  - Mimetic finite differences for elliptic problems
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2009
SP  - 277
EP  - 295
VL  - 43
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2008046/
DO  - 10.1051/m2an:2008046
LA  - en
ID  - M2AN_2009__43_2_277_0
ER  - 
%0 Journal Article
%A Brezzi, Franco
%A Buffa, Annalisa
%A Lipnikov, Konstantin
%T Mimetic finite differences for elliptic problems
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2009
%P 277-295
%V 43
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2008046/
%R 10.1051/m2an:2008046
%G en
%F M2AN_2009__43_2_277_0
Brezzi, Franco; Buffa, Annalisa; Lipnikov, Konstantin. Mimetic finite differences for elliptic problems. ESAIM: Modélisation mathématique et analyse numérique, Volume 43 (2009) no. 2, pp. 277-295. doi : 10.1051/m2an:2008046. http://www.numdam.org/articles/10.1051/m2an:2008046/

[1] P.B. Bochev and J.M. Hyman, Principles of mimetic discretizations of differential operators, IMA Hot Topics Workshop on Compatible Spatial Discretizations 142, D. Arnold, P. Bochev, R. Lehoucq, R. Nicolaides and M. Shashkov Eds., Springer-Verlag (2006). | MR | Zbl

[2] S. Brenner and L. Scott, The mathematical theory of finite element methods. Springer-Verlag, Berlin/Heidelberg (1994). | MR | Zbl

[3] F. Brezzi and A. Buffa, General framework for cochain approximations of differential forms. Technical report, Instituto di Mathematica Applicata a Technologie Informatiche (in preparation).

[4] F. Brezzi, K. Lipnikov and M. Shashkov, Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43 (2005) 1872-1896. | MR | Zbl

[5] F. Brezzi, K. Lipnikov and V. Simoncini, A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Mod. Meth. Appl. Sci. 15 (2005) 1533-1552. | MR | Zbl

[6] F. Brezzi, K. Lipnikov and M. Shashkov, Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes with curved faces. Math. Mod. Meth. Appl. Sci. 16 (2006) 275-297. | MR | Zbl

[7] F. Brezzi, K. Lipnikov, M. Shashkov and V. Simoncini, A new discretization methodology for diffusion problems on generalized polyhedral meshes. Comput. Methods Appl. Mech. Engrg. 196 (2007) 3692-3692. | MR | Zbl

[8] J. Campbell and M. Shashkov, A tensor artificial viscosity using a mimetic finite difference algorithm. J. Comput. Phys. 172 (2001) 739-765. | MR | Zbl

[9] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, New York (1978). | MR | Zbl

[10] M. Dauge, Elliptic boundary value problems on corner domains: smoothness and asymptotics of solutions. Springer-Verlag, Berlin, New York (1988). | MR | Zbl

[11] P. Dvorak, New element lops time off CFD simulations. Mashine Design 78 (2006) 154-155.

[12] S.L. Lyons, R.R. Parashkevov and X.H. Wu, A family of H 1 -conforming finite element spaces for calculations on 3D grids with pinch-outs. Numer. Linear Algebra Appl. 13 (2006) 789-799. | MR | Zbl

[13] L. Margolin, M. Shashkov and P. Smolarkiewicz, A discrete operator calculus for finite difference approximations. Comput. Meth. Appl. Mech. Engrg. 187 (2000) 365-383. | MR | Zbl

[14] P.A. Raviart and J.-M. Thomas, A mixed finite element method for second order elliptic problems, in Mathematical Aspects of the Finite Element Method, I. Galligani and E. Magenes Eds., Springer-Verlag, Berlin-Heilderberg-New York (1977) 292-315. | MR | Zbl

Cited by Sources: