Piecewise linear transformation in diffusive flux discretization

To ensure the discrete maximum principle or solution positivity in finite volume schemes, diffusive flux is sometimes discretized as a conical combination of finite differences. Such a combination may be impossible to construct along material discontinuities using only cell concentration values. This is often resolved by introducing auxiliary node, edge, or face concentration values that are explicitly interpolated from the surrounding cell concentrations. We propose to discretize the diffusive flux after applying a local piecewise linear coordinate transformation that effectively removes the discontinuities. The resulting scheme does not need any auxiliary concentrations and is therefore remarkably simpler, while being second-order accurate under the assumption that the structure of the domain is locally layered.Comment: 11 pages, 1 figures, preprint submitted to Journal of Computational Physic

A Comparison of Consistent Discretizations for Elliptic Problems on Polyhedral Grids

In this work, we review a set of consistent discretizations for second-order elliptic equations, and compare and contrast them with respect to accuracy, monotonicity, and factors affecting their computational cost (degrees of freedom, sparsity, and condition numbers). Our comparisons include the linear and nonlinear TPFA method, multipoint flux-approximation (MPFA-O), mimetic methods, and virtual element methods. We focus on incompressible flow and study the effects of deformed cell geometries and anisotropic permeability.acceptedVersio

A mimetic finite difference method for the Stokes problem with selected edege bubbles

A new mimetic finite difference method for the Stokes problem is proposed and analyzed. The mimetic discretization methodology can be understood as a generalization of the finite element method to meshes with general polygons/polyhedrons. In this paper, the mimetic generalization of the unstable $P_1-P_0$ (and the \u201cconditionally stable\u201d $Q1-P0$) finite element is shown to be fully stable when applied to a large range of polygonal meshes. Moreover, we show how to stabilize the remaining cases by adding a small number of bubble functions to selected mesh edges. A simple strategy for selecting such edges is proposed and verified with numerical experiments