Mimetic finite differences for elliptic problems

Franco Brezzi; Annalisa Buffa; Konstantin Lipnikov

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

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

Abstract

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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.

How to cite

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Brezzi, Franco, Buffa, Annalisa, and Lipnikov, Konstantin. "Mimetic finite differences for elliptic problems." ESAIM: Mathematical Modelling and Numerical Analysis 43.2 (2008): 277-295. <http://eudml.org/doc/194451>.

@article{Brezzi2008,
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. },
author = {Brezzi, Franco, Buffa, Annalisa, Lipnikov, Konstantin},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite differences; polyhedral meshes; diffusion equation; error estimates.; mimetic finite differences; error estimates; convergence; Dirichlet boundary value model problem; numerical comparisons; finite element method},
language = {eng},
month = {12},
number = {2},
pages = {277-295},
publisher = {EDP Sciences},
title = {Mimetic finite differences for elliptic problems},
url = {http://eudml.org/doc/194451},
volume = {43},
year = {2008},
}

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
DA - 2008/12//
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 H1 norm are derived.
LA - eng
KW - Finite differences; polyhedral meshes; diffusion equation; error estimates.; mimetic finite differences; error estimates; convergence; Dirichlet boundary value model problem; numerical comparisons; finite element method
UR - http://eudml.org/doc/194451
ER -

References

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  1. P.B. Bochev and J.M. Hyman, Principles of mimetic discretizations of differential operators, IMA Hot Topics Workshop on Compatible Spatial Discretizations142, D. Arnold, P. Bochev, R. Lehoucq, R. Nicolaides and M. Shashkov Eds., Springer-Verlag (2006).  
  2. S. Brenner and L. Scott, The mathematical theory of finite element methods. Springer-Verlag, Berlin/Heidelberg (1994).  
  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.  
  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.  
  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.  
  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.  
  8. J. Campbell and M. Shashkov, A tensor artificial viscosity using a mimetic finite difference algorithm. J. Comput. Phys.172 (2001) 739–765.  
  9. P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, New York (1978).  
  10. M. Dauge, Elliptic boundary value problems on corner domains: smoothness and asymptotics of solutions. Springer-Verlag, Berlin, New York (1988).  
  11. P. Dvorak, New element lops time off CFD simulations. Mashine Design78 (2006) 154–155.  
  12. S.L. Lyons, R.R. Parashkevov and X.H. Wu, A family of H1-conforming finite element spaces for calculations on 3D grids with pinch-outs. Numer. Linear Algebra Appl.13 (2006) 789–799.  
  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.  
  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.  

Citations in EuDML Documents

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  1. F. Brezzi, Richard S. Falk, L. Donatella Marini, Basic principles of mixed Virtual Element Methods
  2. Robert Eymard, Cindy Guichard, Raphaèle Herbin, Small-stencil 3D schemes for diffusive flows in porous media
  3. Robert Eymard, Cindy Guichard, Raphaèle Herbin, Small-stencil 3D schemes for diffusive flows in porous media
  4. Jérôme Bonelle, Alexandre Ern, Analysis of Compatible Discrete Operator schemes for elliptic problems on polyhedral meshes
  5. Imbunm Kim, Zhongxuan Luo, Zhaoliang Meng, Hyun NAM, Chunjae Park, Dongwoo Sheen, A piecewise P2-nonconforming quadrilateral finite element

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