The Discrete Duality Finite Volume Method for Convection-diffusion Problems

In this paper we extend the discrete duality finite volume (DDFV) formulation to the steady convection-diffusion equation. The discrete gradients defined in DDFV are used to define a cell-based gradient for the control volumes of both the primal and dual meshes, in order to achieve a higher-order accurate numerical flux for the convection term. A priori analysis is carried out to show convergence of the approximation, and a global first-order convergence rate is derived. The theoretical results are confirmed by some numerical experiments.

[1]  Christophe Le Potier,et al.  Schéma volumes finis pour des opérateurs de diffusion fortement anisotropes sur des maillages non structurés , 2005 .

[2]  Raphaèle Herbin,et al.  A finite volume for anisotropic diffusion problems , 2004 .

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

[4]  Yves Coudière,et al.  Convergence rate of a finite volume scheme for the linear convection-diffusion equation on locally refined meshes , 2000 .

[5]  Christophe Le Potier,et al.  Schéma volumes finis monotone pour des opérateurs de diffusion fortement anisotropes sur des maillages de triangles non structurés , 2005 .

[6]  Enrico Bertolazzi,et al.  ON VERTEX RECONSTRUCTIONS FOR CELL-CENTERED FINITE VOLUME APPROXIMATIONS OF 2D ANISOTROPIC DIFFUSION PROBLEMS , 2007 .

[7]  Jérôme Droniou,et al.  Error estimates for the convergence of a finite volume discretization of convection–diffusion equations , 2003, J. Num. Math..

[8]  Timothy J. Barth,et al.  The design and application of upwind schemes on unstructured meshes , 1989 .

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

[10]  Yves Coudière,et al.  Analyse de schemas volumes finis sur maillages non structures pour des problemes lineaires hyperboliques et elliptiques , 1999 .

[11]  F. Hermeline,et al.  Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes , 2003 .

[12]  Robert Eymard,et al.  The finite volume method for Richards equation , 1999 .

[13]  R. Nicolaides,et al.  Covolume Solutions of Three-Dimensional Div-Curl Equations , 1997 .

[14]  Enrico Bertolazzi,et al.  A Second-Order Maximum Principle Preserving Finite Volume Method for Steady Convection-Diffusion Problems , 2005, SIAM J. Numer. Anal..

[15]  R. Herbin,et al.  An Error Estimate for a Nite Volume Scheme for a Diiusion Convection Problem on a Triangular Mesh , 1995 .

[16]  Thierry Gallouët,et al.  A new finite volume scheme for anisotropic diffusion problems on general grids: convergence analysis , 2007 .

[17]  Thierry Gallouët,et al.  Finite volume method , 2010, Scholarpedia.

[18]  Thierry Gallouët,et al.  A cell-centred finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension , 2006 .

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

[20]  S. Nicaise,et al.  A non-conforming finite volume element method of weighted upstream type for the two-dimensional stationary Navier--Stokes system , 2008 .

[21]  Gianmarco Manzini,et al.  Mesh locking effects in the finite volume solution of 2-D anisotropic diffusion equations , 2007, J. Comput. Phys..

[22]  F. Boyer,et al.  Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes , 2007 .

[23]  Robert Eymard,et al.  A mixed finite volume scheme for anisotropic diffusion problems on any grid , 2006, Numerische Mathematik.

[24]  Gianmarco Manzini,et al.  A finite volume method for advection-diffusion problems in convection-dominated regimes , 2008 .

[25]  R. Eymard,et al.  3D Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids , 2008 .

[26]  E. Bertolazzi,et al.  A CELL-CENTERED SECOND-ORDER ACCURATE FINITE VOLUME METHOD FOR CONVECTION–DIFFUSION PROBLEMS ON UNSTRUCTURED MESHES , 2004 .

[27]  F. Hermeline,et al.  A Finite Volume Method for the Approximation of Diffusion Operators on Distorted Meshes , 2000 .

[28]  Pascal Omnes,et al.  A Discrete Duality Finite Volume Approach to Hodge Decomposition and div-curl Problems on Almost Arbitrary Two-Dimensional Meshes , 2007, SIAM J. Numer. Anal..

[29]  Gianmarco Manzini,et al.  Mass-conservative finite volume methods on 2-D unstructured grids for the Richards’ equation , 2004 .

[30]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[31]  R. Nicolaides.  Direct discretization of planar div-curl problems , 1992 .

[32]  Daniil Svyatskiy,et al.  Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes , 2007, J. Comput. Phys..

[33]  S. Agmon.  Lectures on Elliptic Boundary Value Problems , 1965 .

[34]  Roy A. Nicolaides,et al.  Convergence analysis of a covolume scheme for Maxwell's equations in three dimensions , 1998, Math. Comput..

[35]  Enrico Bertolazzi,et al.  Algorithm 817: P2MESH: generic object-oriented interface between 2-D unstructured meshes and FEM/FVM-based PDE solvers , 2002, TOMS.