Two-point stress approximation: A simple and robust finite volume method for linearized (poro-)mechanics and Stokes flow

We provide a unified continuum formulation of linearized mechanics, Stokes' flow and poromechanics in terms of a conservation structure. Starting from this formulation, we construct corresponding simple and robust finite volume discretizations for these physical systems, based only on co-located, cell-centered variables. These discretizations have a minimal discretization stencil, using only the two neighboring cells to a face to calculate numerical stresses and fluxes. We show well-posedness of a weak statement of the continuous formulation in appropriate Hilbert spaces, and identify the appropriate weighted norms for the problem. For the discrete approximations, we prove stability and convergence, both of which are robust in terms of the material parameters. Numerical experiments in 3D support the theoretical results, and provide additional insight into the practical performance of the discretization.