Mixed finite elements for second order elliptic problems in three variables
Numerische Mathematik
On convergence of block-centered finite differences for elliptic-problems
SIAM Journal on Numerical Analysis
Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Enhanced Cell-Centered Finite Differences for Elliptic Equations on General Geometry
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
Quadrilateral H(div) Finite Elements
SIAM Journal on Numerical Analysis
Superconvergence of the Velocity in Mimetic Finite Difference Methods on Quadrilaterals
SIAM Journal on Numerical Analysis
Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes
SIAM Journal on Numerical Analysis
A mortar mimetic finite difference method on non-matching grids
Numerische Mathematik
Robust convergence of multi point flux approximation on rough grids
Numerische Mathematik
A Multipoint Flux Mixed Finite Element Method
SIAM Journal on Numerical Analysis
Superconvergence for Control-Volume Mixed Finite Element Methods on Rectangular Grids
SIAM Journal on Numerical Analysis
Local flux mimetic finite difference methods
Numerische Mathematik
A Composite Mixed Finite Element for Hexahedral Grids
SIAM Journal on Scientific Computing
Error analysis of multipoint flux domain decomposition methods for evolutionary diffusion problems
Journal of Computational Physics
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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-Durán-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.