A priori and a posteriori error bounds for the fully mixed FEM formulation of poroelasticity with stress-dependent permeability

TL;DR

This paper introduces a family of mixed finite element methods for nonlinear poroelasticity with stress-dependent permeability, providing error bounds, adaptive refinement, and demonstrating robustness and accuracy through numerical tests.

Contribution

It develops a novel mixed finite element framework for nonlinear poroelasticity with stress-dependent permeability, including error analysis and adaptive algorithms.

Findings

Method is exactly momentum and mass conservative

Error estimates are reliable and robust in near incompressibility limit

Numerical examples confirm theoretical results

Abstract

We develop a family of mixed finite element methods for a model of nonlinear poroelasticity where, thanks to a rewriting of the constitutive equations, the permeability depends on the total poroelastic stress and on the fluid pressure and therefore we can use the Hellinger--Reissner principle with weakly imposed stress symmetry for Biot's equations. The problem is adequately structured into a coupled system consisting of one saddle-point formulation, one linearised perturbed saddle-point formulation, and two off-diagonal perturbations. This system's unique solvability requires assumptions on regularity and Lipschitz continuity of the inverse permeability, and the analysis follows fixed-point arguments and the Babu\v{s}ka--Brezzi theory. The discrete problem is shown uniquely solvable by applying similar fixed-point and saddle-point techniques as for the continuous case. The method is based on the classical PEERS$_k$ elements, it is exactly momentum and mass conservative, and it is robust with respect to the nearly incompressible as well as vanishing storativity limits. We derive a priori error estimates, we also propose fully computable residual-based a posteriori error indicators, and show that they are reliable and efficient with respect to the natural norms, and robust in the limit of near incompressibility. These a posteriori error estimates are used to drive adaptive mesh refinement. The theoretical analysis is supported and illustrated by several numerical examples in 2D and 3D.