Unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients

Multipoint flux approximation (MPFA) techniques are commonly applied for discretizing the porous media flow equations within the context of finite volume numerical procedures. Although these methods can be applied to heterogeneous, anisotropic systems on generally unstructured grids, the inverse of the resulting linear operator can suffer from a loss of monotonicity at high permeability anisotropy ratios, resulting in spurious oscillations of the pressure solution. The purpose of this paper is to develop a method for optimizing unstructured grids in two and three dimensions such that the monotonicity behavior of the MPFA technique is significantly improved. The method employs anisotropic triangulation and can be readily combined with permeability upscaling procedures. Results are presented for a variety of examples and the technique is shown to perform well on problems involving complex grid point distributions and heterogeneous permeability fields with strong anisotropy ratios (of O(1 0 0)). Oscillation free pressure solutions are achieved in all cases considered, even for examples in which the original grid shows large oscillations.