A Multiscale Mortar Multipoint Flux Mixed Finite Element Method

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. 1991 Mathematics Subject Classification. 65N06, 65N12, 65N15, 65N22, 65N30, 76S05. 07/28/2010. Introduction We consider a second order linear elliptic equation written in a mixed form. Introducing a flux variable, we solve for a scalar function p and a vector function u that satisfy u = −K∇p in Ω, (0.1) ∇ · u = f in Ω, (0.2) p = g on ∂Ω, (0.3)