Weakly imposed symmetry and robust preconditioners for Biot's consolidation model

TL;DR

This paper develops robust preconditioners for finite element methods solving Biot's consolidation model, ensuring stability across parameter variations and discretization refinements, with applications in science and engineering.

Contribution

It introduces preconditioners for Biot's model based on generalized Hellinger-Reissner principles that are robust to parameter changes and discretization refinements.

Findings

Preconditioners are effective across wide parameter ranges.

Preconditioners improve system conditioning in the incompressible limit.

Applicable to finite element methods for poroelasticity and linear elasticity.

Abstract

We discuss the construction of robust preconditioners for finite element approximations of Biot's consolidation model in poroelasticity. More precisely, we study finite element methods based on generalizations of the Hellinger-Reissner principle of linear elasticity, where the stress tensor is one of the unknowns. The Biot model has a number of applications in science, medicine, and engineering. A challenge in many of these applications is that the model parameters range over several orders of magnitude. Therefore, discretization procedures which are well behaved with respect to such variations are needed. The focus of the present paper will be on the construction of preconditioners, such that the preconditioned discrete systems are well-conditioned with respect to variations of the model parameters as well as refinements of the discretization. As a byproduct, we also obtain preconditioners for linear elasticity that are robust in the incompressible limit.