Analysis and preconditioning of parameter-robust finite element methods for Biot's consolidation model

TL;DR

This paper develops and analyzes finite element methods for Biot's consolidation model that remain stable and accurate across a wide range of parameters, with effective preconditioners demonstrated through numerical tests.

Contribution

It introduces parameter-robust finite element discretizations and preconditioners for the Biot model, including error analysis and stability proofs.

Findings

Parameter-robust error estimates achieved for finite element discretizations.

Effective preconditioners based on operator preconditioning are developed.

Numerical experiments confirm theoretical stability and accuracy.

Abstract

In this paper we consider a three-field formulation of the Biot model which has the displacement, the total pressure, and the pore pressure as unknowns. For parameter-robust stability analysis, we first show a priori estimates of the continuous problem with parameter-dependent norms. Then we study finite element discretizations which provide parameter-robust error estimates and preconditioners. For finite element discretizations we consider standard mixed finite element as well as stabilized methods for the Stokes equations, and the complete error analysis of semidiscrete solutions is given. Abstract forms of parameter-robust preconditioners are investigated by the operator preconditioning approach. The theoretical results are illustrated with numerical experiments.