Robust error analysis of coupled mixed methods for Biot's consolidation model

TL;DR

This paper presents a robust a priori error analysis for coupled mixed finite element methods applied to Biot's consolidation model, ensuring stability and accuracy across various material parameters, including nearly incompressible materials.

Contribution

The paper introduces a novel error analysis that guarantees robustness of finite element methods for Biot's model without requiring a positive storage coefficient.

Findings

Error estimates are robust for nearly incompressible materials.

Any pair of stable mixed finite elements can be used.

Numerical experiments confirm theoretical predictions.

Abstract

We study the a priori error analysis of finite element methods for Biot's consolidation model. We consider a formulation which has the stress tensor, the fluid flux, the solid displacement, and the pore pressure as unknowns. Two mixed finite elements, one for linear elasticity and the other for mixed Poisson problems are coupled for spatial discretization, and we show that any pair of stable mixed finite elements is available. The novelty of our analysis is that the error estimates of all the unknowns are robust for material parameters. Specifically, the analysis does not need a uniformly positive storage coefficient, and the error estimates are robust for nearly incompressible materials. Numerical experiments illustrating our theoretical analysis are included.