Robust preconditioners for perturbed saddle-point problems and conservative discretizations of Biot's equations utilizing total pressure

TL;DR

This paper introduces robust preconditioners for perturbed saddle-point problems, specifically targeting mixed formulations of elliptic equations and Biot's equations in poroelasticity, ensuring stability and efficiency across varying material properties.

Contribution

The paper presents new preconditioning techniques that are robust for perturbed saddle-point problems in elliptic and Biot's equations, with detailed stability analysis and numerical validation.

Findings

Preconditioners demonstrate robustness across different material parameters.

Stability of mixed formulations is ensured through weighted space analysis.

Numerical experiments confirm efficiency and robustness of the proposed methods.

Abstract

We develop robust solvers for a class of perturbed saddle-point problems arising in the study of a second-order elliptic equation in mixed form (in terms of flux and potential), and of the four-field formulation of Biot's consolidation problem for linear poroelasticity (using displacement, filtration flux, total pressure and fluid pressure). The stability of the continuous variational mixed problems, which hinges upon using adequately weighted spaces, is addressed in detail; and the efficacy of the proposed preconditioners, as well as their robustness with respect to relevant material properties, is demonstrated through several numerical experiments.