Symmetric Positive Definite General Tensor Discretization Operators on Unstructured and Flow Based Grids

Abstract

S ummaryThe derivation of algebraic flux continuity conditions for full tensor discretization operators has lead to efficient and robust locally conservative flux continuous finite volume methods for determining the discrete velocity field in subsurface reservoirs e .g [1][2][3][4][5][6][7][8] .

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

“…A range of quadrature points is tested and the benefit of using a specific quadrature point is identified, with good convergence results for some challenging cases. However, the formulation of the family of flux-continuous finite-volume schemes in physical space leads to discretization matrices that are not necessarily symmetric in the general case for both quadrilateral and triangular grids [6,7], though it is shown here that the physical-space flux-continuous schemes can still be positive definite subject to ellipticity of the symmetric part of the physical-space tensor.…”

“…More general and improved SPD formulations are presented in [6,7]. These schemes are motivated by the result in [5], where it is proven that an SPD flux-continuous scheme is obtained for q = 1 if each pair of subcell fluxes is defined with respect to a single piecewise constant symmetric elliptic general tensor per subcell, Figure 5, where local numbering refers to the subcells of the control volume.…”

“…which is the source of loss of symmetry in the local subcell flux matrix [6] and consequently loss of symmetry in the global discrete matrix of the physical-space formulation. The effect on discretization is discussed below.…”

“…Also construction of a unique local SPD tensor within the definition of the local subcell flux approximation mirrors the analytical SPD flux property. While both formulations (cell level and subcell level) ensure that the discrete matrix is SPD for q = 1, the subcell formulation first proposed by Edwards [5][6][7] has important advantages over the cell-wise transform-space formulation proposed in [3,10]. By definition a piecewise constant subcell tensor is a superior approximation of the tensor since it allows a finer-scale variation in the tensor geometry, precisely on the sub-scale, resulting in four discrete values compared with a single value at the cell level, for a given quadrilateral cell.…”

“…The first schemes were cell centred and introduced local piecewise constant general-tensor T approximations over the cell or control volume [3,12] for structured grids. The second set of schemes are vertex based and introduce piecewise constant general-tensor approximations over the subcells of each control volume [5][6][7], where the schemes are developed for structured and unstructured grids and are shown to be SPD for any grid (for base quadrature q = 1). These formulations achieve SPD discretization by introduction of an additional local approximation in geometry.…”