Brezzi, Franco; Buffa, Annalisa; Lipnikov, Konstantin
Mimetic finite differences for elliptic problems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 43 (2009) no. 2 , p. 277-295
Zbl 1177.65164 | MR 2512497 | 4 citations dans Numdam
doi : 10.1051/m2an:2008046
URL stable :

Classification:  65N06,  65N12,  65N15,  65N30
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.


[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 2249347 | Zbl 1110.65103

[2] S. Brenner and L. Scott, The mathematical theory of finite element methods. Springer-Verlag, Berlin/Heidelberg (1994). MR 1278258 | Zbl 0804.65101

[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 2192322 | Zbl 1108.65102

[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 2168945 | Zbl 1083.65099

[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 2210091 | Zbl 1094.65111

[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 2339994 | Zbl 1173.76370

[8] J. Campbell and M. Shashkov, A tensor artificial viscosity using a mimetic finite difference algorithm. J. Comput. Phys. 172 (2001) 739-765. MR 1857619 | Zbl 1002.76082

[9] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, New York (1978). MR 520174 | Zbl 0383.65058

[10] M. Dauge, Elliptic boundary value problems on corner domains: smoothness and asymptotics of solutions. Springer-Verlag, Berlin, New York (1988). MR 961439 | Zbl 0668.35001

[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 2266105 | Zbl 1174.65523

[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 1772742 | Zbl 0978.76063

[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 483555 | Zbl 0362.65089