ARBITRARY ORDER NODAL MIMETIC DISCRETIZATIONS OF ELLIPTIC PROBLEMS ON POLYGONAL MESHES WITH ARBITRARY REGULAR SOLUTION

We present a new family of mimetic methods on unstructured polygonal meshes for the diffusion problem in primal form for solution with regularity C fi (›) for any integer fi ‚ 0. These methods are derived from a local consistency condition that is exact for polynomials of degree m = fi + 1. The degrees of freedom are (a) solution and derivative values of various degree at the mesh vertices and (b) solution moments inside polygons. Theoretical results concerning the convergence of the method are briefly summarized and an optimal error estimate is given in a mesh-dependent norm that mimics the energy norm. Numerical experiments confirm the convergence rate that is expected from the theory.

[1]  Lourenço Beirão da Veiga,et al.  A mimetic discretization of the Reissner–Mindlin plate bending problem , 2011, Numerische Mathematik.

[2]  Gianmarco Manzini,et al.  Arbitrary-Order Nodal Mimetic Discretizations of Elliptic Problems on Polygonal Meshes , 2011, SIAM J. Numer. Anal..

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

[4]  J. David Moulton,et al.  A multilevel multiscale mimetic (M3) method for two-phase flows in porous media , 2008, J. Comput. Phys..

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

[6]  Gianmarco Manzini,et al.  Mimetic finite difference method for the Stokes problem on polygonal meshes , 2009, J. Comput. Phys..

[7]  Gianmarco Manzini,et al.  Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems , 2011, J. Comput. Phys..

[8]  M. Shashkov,et al.  The Numerical Solution of Diffusion Problems in Strongly Heterogeneous Non-isotropic Materials , 1997 .

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

[10]  M. Shashkov,et al.  A Local Support-Operators Diffusion Discretization Scheme for Quadrilateralr-zMeshes , 1998 .

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

[12]  Gianmarco Manzini,et al.  A Higher-Order Formulation of the Mimetic Finite Difference Method , 2008, SIAM J. Sci. Comput..

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

[14]  Gianmarco Manzini,et al.  An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems , 2008 .

[15]  Gianmarco Manzini,et al.  Convergence analysis of the high-order mimetic finite difference method , 2009, Numerische Mathematik.

[16]  Gianmarco Manzini,et al.  Error Analysis for a Mimetic Discretization of the Steady Stokes Problem on Polyhedral Meshes , 2010, SIAM J. Numer. Anal..

[17]  Konstantin Lipnikov,et al.  A Mimetic Discretization of the Stokes Problem with Selected Edge Bubbles , 2010, SIAM J. Sci. Comput..

[18]  Stein Krogstad,et al.  Multiscale mixed/mimetic methods on corner-point grids , 2008 .

[19]  Annalisa Buffa,et al.  Mimetic finite differences for elliptic problems , 2009 .

[20]  Gianmarco Manzini,et al.  A unified approach for handling convection terms in finite volumes and mimetic discretization methods for elliptic problems , 2011 .

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

[22]  Gianmarco Manzini,et al.  The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes , 2011, J. Comput. Phys..

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

[24]  Gianmarco Manzini,et al.  Convergence Analysis of the Mimetic Finite Difference Method for Elliptic Problems , 2009, SIAM J. Numer. Anal..

[25]  L. B. D. Veiga,et al.  A Mimetic discretization method for linear elasticity , 2010 .

[26]  Gianmarco Manzini,et al.  Convergence of the mimetic finite difference method for eigenvalue problems in mixed form , 2011 .

[27]  Konstantin Lipnikov,et al.  High-order mimetic finite difference method for diffusion problems on polygonal meshes , 2008, J. Comput. Phys..