We develop a mixed finite element method for single phase flow in porous media that reduces to cell-centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients. Motivated by the multipoint flux approximation method where sub-edge fluxes are introduced, we consider the lowest order Brezzi-Douglas-Marini (BDM) mixed finite element method. A special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell-centered system for the pressures. Theoretical and numerical results indicate second-order convergence for pressures at the cell centers and first-order convergence for sub-edge fluxes. Second-order convergence for edge fluxes is also observed computationally if the grids are sufficiently regular.

Abstract.In this paper, we develop a multiscale mortar multipoint flux mixed finite element method for second order elliptic problems. The equations in the coarse elements (or subdomains) are discretized on a fine grid scale by a multipoint flux mixed finite element method that reduces to cell-centered finite differences on irregular grids. The subdomain grids do not have to match across the interfaces. Continuity of flux between coarse elements is imposed via a mortar finite element space on a coarse grid scale. With an appropriate choice of polynomial degree of the mortar space, we derive optimal order convergence on the fine scale for both the multiscale pressure and velocity, as well as the coarse scale mortar pressure. Some superconvergence results are also derived. The algebraic system is reduced via a non-overlapping domain decomposition to a coarse scale mortar interface problem that is solved using a multiscale flux basis. Numerical experiments are presented to confirm the theory and illustrate the efficiency and flexibility of the method.Mathematics Subject Classification. 65N06, 65N12, 65N15, 65N22, 65N30, 76S05.

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