Quasi-positive families of continuous Darcy-flux finite volume schemes on structured and unstructured grids

New families of flux-continuous control-volume distributed finite volume schemes are presented for the general full-tensor pressure equation arising in porous media and formulated for structured and unstructured grids. These schemes offer the practical advantage of being flux-continuous while only depending on one degree of freedom per control-volume, unlike rival approximations such as the Mixed Finite Element method. M -matrix bounds are presented, quasi QM-matrices are defined and an optimal quadrilateral scheme is identified. Anisotropy favoring triangulation is also shown to yield an optimal scheme. The new schemes prove to be relatively robust for the cases tested, including strongly anisotropic full tensor fields. Strong oscillations encountered with the earlier formulations, are removed or minimized.