Inf-sup stable discretization of the quasi-static Biot's equations in poroelasticity

TL;DR

This paper introduces a new stable discretization method for Biot's equations in poroelasticity, ensuring robustness and optimal error decay, based on inf-sup theory and finite element discretization.

Contribution

It develops an inf-sup stable, fully discretized scheme for Biot's equations using a four-field formulation and proves its stability and error decay properties.

Findings

Establishes inf-sup stability and quasi-optimality of the discretization.

Provides error decay estimates for smooth solutions.

Ensures robustness of constants with respect to material parameters.

Abstract

We propose a new full discretization of the Biot's equations in poroelasticity. The construction is driven by the inf-sup theory, which we recently developed. It builds upon the four-field formulation of the equations obtained by introducing the total pressure and the total fluid content. We discretize in space with Lagrange finite elements and in time with backward Euler. We establish inf-sup stability and quasi-optimality of the proposed discretization, with robust constants with respect to all material parameters. We further construct an interpolant showing how the error decays for smooth solutions.