Quasi M-Matrix Multifamily Continuous Darcy-Flux Approximations with Full Pressure Support on Structured and Unstructured Grids in Three Dimensions

Authors:
Michael G. Edwards;Hongwen Zheng
Affiliations:
m.g.edwards@swansea.ac.uk;hongwen.zheng@cmgl.ca
Venue:
SIAM Journal on Scientific Computing
Year:
2011

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Abstract

New multifamilies of flux-continuous vertex centered finite-volume methods are presented for the full-tensor pressure equation with general discontinuous coefficients for any cell type in three dimensions. The new schemes are flux continuous with full pressure support over each subcell with continuous pressure imposed across each control-volume subface, in contrast to earlier formulations. Full pressure continuity across subfaces leads to a quasi-positive formulation that minimizes spurious oscillations in discrete pressure solutions for strongly anisotropic full-tensor fields. The multifamily formulation permits maximum flexibility in quadrature, yielding improved solution resolution. The earlier methods are pointwise continuous in pressure and flux with tetrahedral pressure support, which leads to a more limited quadrature range, which is shown to cause spurious oscillations in the solution for strong full-tensor fields. An M-matrix analysis of the three-dimensional schemes identifies bounding limits for the schemes to possess a local discrete maximum principle. Conditions for the schemes to be positive definite are also derived.