Abstract
We present a new mixed finite element for hexahedra, conforming in H(div). The new velocity space is similar in construction to that of BDDF of order one, but has four degrees of freedom per face instead of three. The natural degrees of freedom defining the velocity field are the components of the velocity normal to each face at the vertices of that face. We apply the new element to the numerical solution of second order elliptic equations with full tensor coefficients. We prove convergence of the method and provide optimal error estimates for the scalar and vector fields. We use an appropriate numerical quadrature to localize the mixed finite element method into a cell-centered finite difference method. For diagonal tensor coefficients, we recover the usual 7-point stencil. For full tensors, the method leads to a 27-point stencil of multipoint flux approximation (MPFA) methods—specifically the well-known O-method. Our construction allows us to prove, for the first time, convergence of the MFPA O-method on three-dimensional rectangular hexahedra. The results of the error analysis are confirmed by numerical experiments. Although our analysis is restricted to rectangular parallelepipeds, our implementation allows for distorted elements and unstructured grids.
