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Hybrid and multi-point formulations of the lowest-order mixed methods for Darcy’s flow on triangles. (English) Zbl 1149.76030

Summary: 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.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
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[1] Durlofsky, Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities, Water Resources Research 21 pp 965– (1994)
[2] Darlow, Mixed finite element method for miscible displacement problems in porous media, Society of Petroleum Engineers Journal 24 pp 391– (1984)
[3] Brezzi, Mixed and Hybrid Finite Element Methods (1991) · Zbl 0788.73002
[4] Bergamaschi L, Mantica S, Saleri F. Mixed finite element approximation of Darcy’s law in porous media. Technical Report, CRS4, Cagliari, Italy, 1994.
[5] Mosé, Application of the mixed hybrid finite element approximation in a groundwater flow model: luxury or necessity?, Water Resources Research 30 pp 3001– (1994)
[6] Younes, A new formulation of the mixed finite element method for solving elliptic and parabolic PDE with triangular elements, Journal of Computational Physics 149 pp 148– (1999) · Zbl 0923.65064
[7] Ackerer, Modeling variable density flow and solute transport in porous medium: 1. Numerical model and verification, Transport in Porous Media 35 pp 345– (1999)
[8] Chavent, On the finite volume reformulation of the mixed finite element method for elliptic and parabolic PDE on triangles, Computer Methods in Applied Mechanics and Engineering 192 pp 655– (2003) · Zbl 1091.76520
[9] Younes, A new mass lumping scheme for the mixed hybrid finite element method, International Journal for Numerical Methods in Engineering 67 pp 89– (2006) · Zbl 1114.65116
[10] Chavent, Mathematical Models and Finite Elements for Reservoir Simulation (1986) · Zbl 0603.76101
[11] Roberts, Mixed and Hybrid Methods II (1989)
[12] Russel, The Mathematics of Reservoir Simulation (1983)
[13] Weiser, On convergence of block-centered finite differences for elliptic problems, SIAM Journal on Numerical Analysis 25 pp 351– (1988) · Zbl 0644.65062
[14] Baranger, Application de la théorie des éléments finis mixtes à l’étude d’une classe de schémas aux volumes-différences finis pour les problèmes elliptiques, Comptes Rendus de l’Academie des Sciences Paris, Série I 319 pp 401– (1994)
[15] Arbogast T, Kennan PT. Mixed finite element methods as finite difference methods for solving elliptic equations on triangular elements. Technical Report 93-53, Department of Computational and Applied Mathematics, Rice University, 1984.
[16] Younes, From mixed finite elements to finite volumes for elliptic PDE in 2 and 3 dimensions, International Journal for Numerical Methods in Engineering 59 pp 365– (2004)
[17] Aavatsmark I, Barkve T, Bøe Ø, Mannseth T. Discretization on non-orthogonal, curvilinear grids for multiphase flow. In Proceedings of 4th European Conference on the Mathematics of Oil Recovery, Christie MA, Farmer CL, Guillon O, Heinmann ZE (eds). Norway, 1994. · Zbl 0859.76048
[18] Edwards MG, Rogers CF. A flux continuous scheme for the full tensor pressure equation. Proceedings of the 4th European Conference on the Mathematics of Oil Recovery, Røros, 1994.
[19] Aavatsmark, An introduction to multipoint flux approximations for quadrilateral grids, Journal of Computational Geosciences 6 pp 404– (2002) · Zbl 1094.76550
[20] Edwards, Finite volume discretization with imposed flux continuity for the general tensor pressure equation, Journal of Computational Geosciences 2 pp 259– (1998) · Zbl 0945.76049
[21] Vohralik, Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes, Mathematical Modelling and Numerical Analysis 40 pp 367– (2006)
[22] Klausen, Convergence of multipoint flux approximations on quadrilateral grids, Numerical Methods for Partial Differential Equations 22 pp 1438– (2006) · Zbl 1106.76043
[23] Klausen, Robust convergence of multi-point flux approximations on rough grids, Numerische Mathematik 104 pp 317– (2006) · Zbl 1102.76036
[24] Wheeler, Compatible Spatial Discretizations pp 189– (2006)
[25] Wheeler, A multipoint flux mixed finite element method, SIAM Journal on Numerical Analysis 44 pp 2082– (2006) · Zbl 1121.76040
[26] Aavatsmark I, Eigestad G, Klausen RA, Wheeler MF, Yotov I. Convergence of a symmetric MPFA method on quadrilateral grids. Technical Report TR-MATH 05-14, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 2005. · Zbl 1128.65093
[27] Fraeijs de Veubeke BX. Displacement and equilibrium models in the finite element method. In Stress Analysis, Zienkiewicz OC, Hollister G (eds). New York, 1965.
[28] Brezzi, Two families of mixed finite elements for second order elliptic problems, Numerische Mathematik 47 pp 217– (1985) · Zbl 0599.65072
[29] Brezzi, Finite Elements Methods: 1970 and Beyond (2004)
[30] Pal, Convergence study of a family of flux-continuous, finite-volume schemes for the general tensor pressure equation, International Journal for Numerical Methods in Fluids 51 pp 1177– (2006) · Zbl 1108.76046
[31] Le Potier, Schéma volumes finis pour des opérateurs de diffusion fortement anisotropes sur des maillages non structures, Comptes Rendu Mathematique 12 pp 921– (2005)
[32] Eisenstat, Efficient implementation of a class of preconditioned conjugate gradient methods, SIAM Journal on Scientific and Statistical Computing 2 pp 1– (1981) · Zbl 0474.65020
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