An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems

We present an a posteriori error indicator for the mimetic finite difference approximation of elliptic problems in the mixed form. We show that this estimator is reliable and efficient with respect to an energy-type error comprising both flux and pressure. Its performance is investigated by numerically solving the diffusion equation on computational domains with different shapes, different permeability tensors, and different types of computational meshes. Copyright © 2008 John Wiley & Sons, Ltd.

[1]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis: Oden/A Posteriori , 2000 .

[2]  Gianmarco Manzini,et al.  Flux reconstruction and solution post-processing in mimetic finite difference methods , 2008 .

[3]  F. Brezzi,et al.  A FAMILY OF MIMETIC FINITE DIFFERENCE METHODS ON POLYGONAL AND POLYHEDRAL MESHES , 2005 .

[4]  M. Shashkov,et al.  A new discretization methodology for diffusion problems on generalized polyhedral meshes , 2007 .

[5]  M. Shashkov,et al.  The Orthogonal Decomposition Theorems for Mimetic Finite Difference Methods , 1999 .

[6]  Konstantin Lipnikov,et al.  Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes , 2005, SIAM J. Numer. Anal..

[7]  Martin Vohralík,et al.  A Posteriori Error Estimates for Lowest-Order Mixed Finite Element Discretizations of Convection-Diffusion-Reaction Equations , 2007, SIAM J. Numer. Anal..

[8]  M. Shashkov,et al.  Mimetic Finite Difference Methods for Maxwell's Equations and the Equations of Magnetic Diffusion - Abstract , 2001 .

[9]  Lourenço Beirão da Veiga,et al.  A residual based error estimator for the Mimetic Finite Difference method , 2007, Numerische Mathematik.

[10]  Pascal Omnes,et al.  A FINITE VOLUME METHOD FOR THE LAPLACE EQUATION ON ALMOST ARBITRARY TWO-DIMENSIONAL GRIDS , 2005 .

[11]  Carsten Carstensen,et al.  A posteriori error estimate for the mixed finite element method , 1997, Math. Comput..

[12]  M. Shashkov,et al.  CONVERGENCE OF MIMETIC FINITE DIFFERENCE METHOD FOR DIFFUSION PROBLEMS ON POLYHEDRAL MESHES WITH CURVED FACES , 2006 .

[13]  J. David Moulton,et al.  Convergence of mimetic finite difference discretizations of the diffusion equation , 2001, J. Num. Math..

[14]  M. Shashkov,et al.  The mimetic finite difference method on polygonal meshes for diffusion-type problems , 2004 .

[15]  Mikhail Shashkov,et al.  Approximation of boundary conditions for mimetic finite-difference methods , 1998 .

[16]  Rolf Stenberg,et al.  Energy norm a posteriori error estimates for mixed finite element methods , 2006, Math. Comput..

[17]  Mikhail Shashkov,et al.  A tensor artificial viscosity using a mimetic finite difference algorithm , 2001 .

[18]  M. Shashkov,et al.  Adjoint operators for the natural discretizations of the divergence gradient and curl on logically rectangular grids , 1997 .

[19]  M. Shashkov,et al.  The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes , 2006 .

[20]  Mikhail Shashkov,et al.  Solving Diffusion Equations with Rough Coefficients in Rough Grids , 1996 .

[21]  Mikhail Shashkov,et al.  Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes , 2004 .

[22]  B. Rivière,et al.  Part II. Discontinuous Galerkin method applied to a single phase flow in porous media , 2000 .

[23]  Mary F. Wheeler,et al.  Superconvergence of the Velocity in Mimetic Finite Difference Methods on Quadrilaterals , 2005, SIAM J. Numer. Anal..

[24]  M. Shashkov,et al.  Mimetic Finite Difference Methods for Diffusion Equations , 2002 .

[25]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .