A Multipoint Flux Mixed Finite Element Method on Hexahedra

Abstract

We develop a mixed finite element method for elliptic problems on hexahedral grids that reduces to cell-centered finite differences. The paper is an extension of our earlier paper for quadrilateral and simplicial grids [M. F. Wheeler and I. Yotov, SIAM J. Numer. Anal., 44 (2006), pp. 2082-2106]. The construction is motivated by the multipoint flux approximation method, and it is based on an enhancement of the lowest order Brezzi-Douglas-Duran-Fortin (BDDF) mixed finite element spaces on hexahedra. In particular, there are four fluxes per face, one associated with each vertex. 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 first-order convergence for pressures and subface fluxes on sufficiently regular grids, as well as second-order convergence for pressures at the cell centers. Second-order convergence for face fluxes is also observed computationally.