Positive‐definiteq‐families of continuous subcell Darcy‐flux CVD(MPFA) finite‐volume schemes and the mixed finite element method

Abstract

SUMMARYA new family of locally conservative cell-centred flux-continuous schemes is presented for solving the porous media general-tensor pressure equation. A general geometry-permeability tensor approximation is introduced that is piecewise constant over the subcells of the control volumes and ensures that the local discrete general tensor is elliptic. A family of control-volume distributed subcell flux-continuous schemes are defined in terms of the quadrature parametrization q (Multigrid Methods. Birkhauser: Basel, 1993; Proceedings of the 4th European Conference on the Mathematics of Oil Recovery, Norway, June 1994;Comput. Geosci. 1998; 2:259-290), where the local position of flux continuity defines the quadrature point and each particular scheme. The subcell tensor approximation ensures that a symmetric positive-definite (SPD) discretization matrix is obtained for the base member (q = 1) of the formulation. The physical-space schemes are shown to be non-symmetric for general quadrilateral cells. Conditions for discrete ellipticity of the non-symmetric schemes are derived with respect to the local symmetric part of the tensor. The relationship with the mixed finite element method is given for both the physical-space and subcell-space q-families of schemes. M-matrix monotonicity conditions for these schemes are summarized. A numerical convergence study of the schemes shows that while the physical-space schemes are the most accurate, the subcell tensor approximation reduces solution errors when compared with earlier cell-wise constant tensor schemes and that subcell tensor approximation using the control-volume face geometry yields the best SPD scheme results. A particular quadrature point is found to improve numerical convergence of the subcell schemes for the cases tested.

Multi‐dimensional wave‐oriented upwind schemes with reduced cross‐wind diffusion for flow in porous media

“…Full tensors arise when the local coordinate system is non-aligned with the principal axes of the tensor [33], which can occur with both structured grids and general unstructured grids [34][35][36][37][38]. Full tensors can also arise in upscaling when cross-flow effects are included.…”

Non‐linear flux‐splitting schemes with imposed discrete maximum principle for elliptic equations with highly anisotropic coefficients

“…Thus any scheme applicable to a full tensor also applies to non-K -Orthogonal grids. Note that T 11 , T 22 0 and the ellipticity of T follows from Equations (3) and (7). Full tensors can arise from upscaling, and orientation of grid and permeability field.…”

“…Symmetric positive definite (SPD) conditions are derived in [2-5, 7, 21]. While physical space formulations do not possess symmetric discretization matrices for arbitrary quadrilaterals, a physical space SPD scheme is presented in [21] for cell-centred triangular grids and conditions for positive definite formulations are presented in [7,21]. Coupling of the flux-continuous schemes with higher-order convective flux approximations applied to multi-phase flow on structured and unstructured grids in two and three dimensions is presented in [22,23], respectively.…”

Comparison of cell- and vertex-centered discretization methods for flow in a two-dimensional discrete-fracture–matrix system

“…For an overview of different MPFA schemes see [2,1,3,28,67], and references therein. The letter "O" comes from the shape of the polylines connecting the involved grid points in a cell-stensil [2].…”

“…In theory, the effects of each parameter on the condition number could be determined by looking at the coefficient matrix A and how parameters enter the matrix, and relations between the different elements. However, using this approach we were only able to discover the general relation in Equation (28); the form of the function f G M in that equation could only be determined from extensive numerical simulations for given fracture configuration G and selected values of the parameters α, β and γ.…”