Accurate discretization of poroelasticity without Darcy stability: Stokes–Biot stability revisited

Abstract

In this manuscript we focus on the question: what is the correct notion of Stokes–Biot stability? Stokes–Biot stable discretizations have been introduced, independently by several authors, as a means of discretizing Biot’s equations of poroelasticity; such schemes retain their stability and convergence properties, with respect to appropriately defined norms, in the context of a vanishing storage coefficient and a vanishing hydraulic conductivity. The basic premise of a Stokes–Biot stable discretization is: one part Stokes stability and one part mixed Darcy stability. In this manuscript we remark on the observation that the latter condition can be generalized to a wider class of discrete spaces. In particular: a parameter-uniform inf-sup condition for a mixed Darcy sub-problem is not strictly necessary to retain the practical advantages currently enjoyed by the class of Stokes–Biot stable Euler–Galerkin discretization schemes.