Hybrid and multi-point formulations of the lowest-order mixed methods for Darcy's flow on triangles

Mixed finite element (MFE) and multipoint flux approximation (MPFA) methods have similar properties and are well suited for the resolution of Darcy's flow on anisotropic and heterogeneous domains. In this work, the link between hybrid and MPFA formulations is shown algebraically for the lowest order mixed methods of Raviart–Thomas (RT0) and Brezzi–Douglas–Marini (BDM1) on triangles. The efficiency of the four mixed formulations (Hybrid_RT0, MPFA_RT0, Hybrid_BDM1 and MPFA_BDM1) is investigated on high anisotropic and heterogeneous media and for unstructured triangular discretizations. Numerical experiments show that the MPFA_BDM1 formulation outperforms both Hybrid_RT0 and Hybrid_BDM1 in the case of anisotropic domains and highly unstructured meshes. Copyright © 2008 John Wiley & Sons, Ltd.

[1]  L. D. Marini,et al.  Two families of mixed finite elements for second order elliptic problems , 1985 .

[2]  A. Weiser,et al.  On convergence of block-centered finite differences for elliptic-problems , 1988 .

[3]  G. Chavent,et al.  A New Formulation of the Mixed Finite Element Method for Solving Elliptic and Parabolic PDE with Triangular Elements , 1999 .

[4]  Ragnar Winther,et al.  Robust convergence of multi point flux approximation on rough grids , 2006, Numerische Mathematik.

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

[6]  Michael G. Edwards,et al.  A Flux Continuous Scheme for the Full Tensor Pressure Equation , 1994 .

[7]  J. Baranger,et al.  APPLICATION DE LA THEORIE DES ELEMENTS FINIS MIXTES A L'ETUDE D'UNE CLASSEDE SCHEMAS AUX VOLUMES DIFFERENCES FINIS POUR LES PROBLEMES ELLIPTIQUES , 1994 .

[8]  S. Eisenstat.  Efficient Implementation of a Class of Preconditioned Conjugate Gradient Methods , 1981 .

[9]  Michael G. Edwards,et al.  Finite volume discretization with imposed flux continuity for the general tensor pressure equation , 1998 .

[10]  M. Vohralík.  Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes , 2006 .

[11]  Ragnar Winther,et al.  Convergence of multipoint flux approximations on quadrilateral grids , 2006 .

[12]  P. Ackerer,et al.  A new mass lumping scheme for the mixed hybrid finite element method , 2006 .

[13]  Todd Arbogast,et al.  Mixed Finite Element Methods as Finite Difference Methods for Solving Elliptic Equations on Triangular Elements , 1993 .

[14]  G. Chavent,et al.  Application of the mixed hybrid finite element approximation in a groundwater flow model: Luxury or necessity? , 1994 .

[15]  Mayur Pal,et al.  Convergence study of a family of flux‐continuous, finite‐volume schemes for the general tensor pressure equation , 2006 .

[16]  L. Durlofsky.  Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities , 1994 .

[17]  Richard E. Ewing,et al.  Mixed Finite Element Method for Miscible Displacement Problems in Porous Media , 1984 .

[18]  Mary F. Wheeler,et al.  A Cell-Centered Finite Difference Method on Quadrilaterals , 2006 .

[19]  I. Aavatsmark,et al.  An Introduction to Multipoint Flux Approximations for Quadrilateral Grids , 2002 .

[20]  I. Aavatsmark,et al.  Discretization on Non-Orthogonal, Curvilinear Grids for Multi-Phase Flow , 1994 .

[21]  G. Chavent,et al.  On the finite volume reformulation of the mixed finite element method for elliptic and parabolic PDE on triangles , 2003 .

[22]  P. Ackerer,et al.  Modeling Variable Density Flow and Solute Transport in Porous Medium: 1. Numerical Model and Verification , 1999 .

[23]  Mary F. Wheeler,et al.  A Multipoint Flux Mixed Finite Element Method , 2006, SIAM J. Numer. Anal..

[24]  G. Chavent,et al.  From mixed finite elements to finite volumes for elliptic PDEs in two and three dimensions , 2004 .