Title:          A multipoint flux mixed finite element method on hexahedra


Authors:        R. Ingram, M. F. Wheeler, and I. Yotov


Source:         University of Pittsburgh Tech Report TR-MATH 09-24


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 [SIAM J. Numer. Anal., Vol. 55,
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--Dur\'an--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 sub-face 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.