A Family of Multipoint Flux Mixed Finite Element Methods for Elliptic Problems on General Grids

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Abstract

In this paper, we discuss a family of multipoint flux mixed finite element (MFMFE) methods on simplicial, quadrilateral, hexahedral, and triangular-prismatic grids. The MFMFE methods are locally conservative with continuous normal fluxes, since they are developed within a variational framework as mixed finite element methods with special approximating spaces and quadrature rules. The latter allows for local flux elimination giving a cell-centered system for the scalar variable. We study two versions of the method: with a symmetric quadrature rule on smooth grids and a non-symmetric quadrature rule on rough grids. Theoretical and numerical results demonstrate first order convergence for problems with full-tensor coeffcients. Second order superconvergence is observed on smooth grids.

Keywords

mixed finite element
multipoint flux approximation
cell-centered finite difference
full tensor
simplices
quadrilaterals
hexahedra
triangular prisms

Cited by (0)

1

partially supported by the NSF-CDI under contract number DMS 0835745, the DOE grant DE-FG02-04ER25617, and the Center for Frontiers of Subsurface Energy Security under Contract No. DE-SC0001114.

2

supported by Award No. KUS-F1-032-04, made by King Abdullah University of Science and Technology (KAUST).

3

partially supported by the DOE grant DE-FG02-04ER25618, the NSF grant DMS 0813901, and the J. Tinsley Oden Faculty Fellowship, ICES, The University of Texas at Austin.