Finite volume cell centered discretizations of multiphase porous media flow are very popular in the oil industry for their low computational cost and their ability to incorporate complex nonlinear closure laws and physics. However, the trend towards a more accurate description of the geometry and of the porous medium requires to dispose of flux approximations handling general polyhedral meshes and full diffusion tensors. The convergence on general meshes as well as the robustness with respect to anisotropy and heterogeneity of the diffusion tensor should come at a reasonable computational cost. An important property on which the analysis relies is coercivity, which ensures the stability of the scheme and allows to prove convergence of consistent discretizations. Meeting all these requirements is still a challenge and this is an active field of research. The L-method, which has been recently introduced by Ivar Aavatsmark and co-workers, displays enhanced monotonicity properties on distorted meshes and for anisotropic diffusion tensors. In this work, we present a generalization of the L-method based on a discrete variational framework and on constant gradient reconstructions. The coercivity of the discretization requires a local condition depending both on the mesh and on the diffusion tensor to be satisfied. The monotonicity and convergence properties of the scheme are assessed on challenging single-phase problems with distorted meshes and anisotropic and heterogeneous diffusion tensors. The proposed method is compared with the standard O- and L-methods as well as with an unconditionally symmetric, coercive cell centered finite volume scheme using a larger stencil.


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