A generalized Mimetic Finite Difference method and Two-Point Flux schemes over Voronoi diagrams

¹ Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 East 24th Street, Austin, TX 78712, USA.
ohinai@gmail.com; mfw@ices.utexas.edu
² Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA0.
yotov@math.pitt.edu

Received: 5 September 2015
Revised: 13 April 2016
Accepted: 5 May 2016

Abstract

We develop a generalization of the mimetic finite difference (MFD) method for second order elliptic problems that extends the family of convergent schemes to include two-point flux approximation (TPFA) methods over general Voronoi meshes, which are known to satisfy the discrete maximum principle. The method satisfies a modified consistency condition, which utilizes element and face weighting functions. This results in shifting the points on the elements and faces where the pressure and the flux are most accurately approximated. The flux bilinear form is non-symmetric in general, although it reduces to a symmetric form in the case of TPFA. It can be defined as the L²-inner product of vectors in two H(Ω;div) discrete spaces, which are constructed via suitable lifting operators. A specific construction of such lifting operators is presented on rectangles. We note that a different choice is made for test and trial spaces, therefore the method can be viewed as a H(Ω;div)-conforming Petrov–Galerkin Mixed Finite Element method. We prove first-order convergence in pressure and flux, and superconvergence of the pressure under further restrictions. We present numerical results that support the theory.