Inf-sup theory for the quasi-static Biot's equations in poroelasticity

TL;DR

This paper applies inf-sup theory to analyze the well-posedness and stability of the quasi-static Biot's equations in poroelasticity, introducing a four-field formulation to establish robust existence, uniqueness, and regularity results.

Contribution

It introduces an equivalent four-field formulation for Biot's equations and proves robust stability, existence, and regularity results, aiding in the development of accurate discretizations.

Findings

Established existence and uniqueness of solutions.

Proved stability estimates independent of material parameters.

Showed additional regularity of solutions under data regularity.

Abstract

We analyze the two-field formulation of the quasi-static Biot's equations in bounded domains by means of the inf-sup theory. For this purpose, we exploit an equivalent four-field formulation of the equations, introducing the so-called total pressure and total fluid content as independent variables. We establish existence, uniqueness and stability of the solution. Our stability estimate is two-sided and robust, meaning that the regularity established for the solution matches the regularity requirements for the data and the involved constants are independent of all material parameters. We prove also that additional regularity in space of the data implies, in some cases, corresponding additional regularity in space of the solution. These results are instrumental to the design and the analysis of discretizations enjoying accurate stability and error estimates.