Mimetic finite differences for elliptic problems

Franco Brezzi; Annalisa Buffa; Konstantin Lipnikov

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2009)

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

Abstract

top
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.

How to cite

top

Brezzi, 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. [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. [2] S. Brenner and L. Scott, The mathematical theory of finite element methods. Springer-Verlag, Berlin/Heidelberg (1994). Zbl0804.65101MR1278258
  3. [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. [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. [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. [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. [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. [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. [9] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, New York (1978). Zbl0383.65058MR520174
  10. [10] M. Dauge, Elliptic boundary value problems on corner domains: smoothness and asymptotics of solutions. Springer-Verlag, Berlin, New York (1988). Zbl0668.35001MR961439
  11. [11] P. Dvorak, New element lops time off CFD simulations. Mashine Design 78 (2006) 154–155. 
  12. [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. [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. [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 ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.