Accurate Discretization Of Poroelasticity Without Darcy Stability -- Stokes-Biot Stability Revisited

TL;DR

This paper revisits the concept of Stokes-Biot stability in discretizing poroelasticity equations, proposing a broader class of stable schemes that do not require strict Darcy stability conditions.

Contribution

It generalizes the notion of Darcy stability in Stokes-Biot discretizations, allowing for more flexible and potentially more efficient numerical schemes.

Findings

A parameter-uniform inf-sup condition is not necessary for stability.

Broader classes of discrete spaces can achieve Stokes-Biot stability.

Enhanced understanding of stability conditions improves discretization methods.

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.