# Mimetic finite differences for elliptic problems

Franco Brezzi; Annalisa Buffa; Konstantin Lipnikov

- Volume: 43, Issue: 2, page 277-295
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topBrezzi, Franco, Buffa, Annalisa, and Lipnikov, Konstantin. "Mimetic finite differences for elliptic problems." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 43.2 (2009): 277-295. <http://eudml.org/doc/245118>.

@article{Brezzi2009,

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 $H^1$ norm are derived.},

author = {Brezzi, Franco, Buffa, Annalisa, Lipnikov, Konstantin},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {finite differences; polyhedral meshes; diffusion equation; error estimates; mimetic finite differences; convergence; Dirichlet boundary value model problem; numerical comparisons; finite element method},

language = {eng},

number = {2},

pages = {277-295},

publisher = {EDP-Sciences},

title = {Mimetic finite differences for elliptic problems},

url = {http://eudml.org/doc/245118},

volume = {43},

year = {2009},

}

TY - JOUR

AU - Brezzi, Franco

AU - Buffa, Annalisa

AU - Lipnikov, Konstantin

TI - Mimetic finite differences for elliptic problems

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2009

PB - EDP-Sciences

VL - 43

IS - 2

SP - 277

EP - 295

AB - 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.

LA - eng

KW - finite differences; polyhedral meshes; diffusion equation; error estimates; mimetic finite differences; convergence; Dirichlet boundary value model problem; numerical comparisons; finite element method

UR - http://eudml.org/doc/245118

ER -

## References

top- [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). Zbl1110.65103MR2249347
- [2] S. Brenner and L. Scott, The mathematical theory of finite element methods. Springer-Verlag, Berlin/Heidelberg (1994). Zbl0804.65101MR1278258
- [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). Zbl1191.78056
- [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. Zbl1108.65102MR2192322
- [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. Zbl1083.65099MR2168945
- [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. Zbl1094.65111MR2210091
- [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. Zbl1173.76370MR2339994
- [8] J. Campbell and M. Shashkov, A tensor artificial viscosity using a mimetic finite difference algorithm. J. Comput. Phys. 172 (2001) 739–765. Zbl1002.76082MR1857619
- [9] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, New York (1978). Zbl0383.65058MR520174
- [10] M. Dauge, Elliptic boundary value problems on corner domains: smoothness and asymptotics of solutions. Springer-Verlag, Berlin, New York (1988). Zbl0668.35001MR961439
- [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. Zbl1174.65523MR2266105
- [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. Zbl0978.76063MR1772742
- [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. Zbl0362.65089MR483555

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.