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Mimetic finite differences for elliptic problems

Published online by Cambridge University Press:  05 December 2008

Franco Brezzi ,
Annalisa Buffa  and
Konstantin Lipnikov
Franco Brezzi
Affiliation:
Istituto Universitario di Studi Superiori, Pavia, Italy. [email protected]
Annalisa Buffa
Affiliation:
Instituto di Matematica Applicata e Tecnologie Informatiche, Pavia, Italy. [email protected]
Konstantin Lipnikov
Affiliation:
Los Alamos National Laboratory, Theoretical Division, MS B284, Los Alamos, NM, 87545, USA. [email protected]
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Abstract

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 H1 norm are derived.

Keywords

Finite differencespolyhedral meshesdiffusion equationerror estimates.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 2 , March 2009 , pp. 277 - 295
Copyright
© EDP Sciences, SMAI, 2009

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References

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).
S. Brenner and L. Scott, The mathematical theory of finite element methods. Springer-Verlag, Berlin/Heidelberg (1994).
F. Brezzi and A. Buffa, General framework for cochain approximations of differential forms. Technical report, Instituto di Mathematica Applicata a Technologie Informatiche (in preparation).
Brezzi, F., Lipnikov, K. and Shashkov, M., Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43 (2005) 1872–1896. CrossRef
Brezzi, F., Lipnikov, K. and Simoncini, V., A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Mod. Meth. Appl. Sci. 15 (2005) 1533–1552. CrossRef
Brezzi, F., Lipnikov, K. and Shashkov, M., Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes with curved faces. Math. Mod. Meth. Appl. Sci. 16 (2006) 275–297. CrossRef
Brezzi, F., Lipnikov, K., Shashkov, M. and Simoncini, V., A new discretization methodology for diffusion problems on generalized polyhedral meshes. Comput. Methods Appl. Mech. Engrg. 196 (2007) 3692–3692. CrossRef
Campbell, J. and Shashkov, M., A tensor artificial viscosity using a mimetic finite difference algorithm. J. Comput. Phys. 172 (2001) 739–765. CrossRef
P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, New York (1978).
M. Dauge, Elliptic boundary value problems on corner domains: smoothness and asymptotics of solutions. Springer-Verlag, Berlin, New York (1988).
Dvorak, P., New element lops time off CFD simulations. Mashine Design 78 (2006) 154–155.
Lyons, S.L., Parashkevov, R.R. and Wu, X.H., 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. CrossRef
Margolin, L., Shashkov, M. and Smolarkiewicz, P., A discrete operator calculus for finite difference approximations. Comput. Meth. Appl. Mech. Engrg. 187 (2000) 365–383. CrossRef
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.