Robust iterative schemes for non-linear poromechanics

TL;DR

This paper introduces two convergent iterative schemes for solving a non-linear extension of Biot's poromechanics model, enabling stable and efficient numerical simulations of coupled fluid flow and deformation.

Contribution

It extends classical linear poromechanics models to non-linear cases and provides rigorous convergence proofs for the proposed iterative schemes.

Findings

Both schemes converge under specified conditions.

Numerical examples confirm theoretical convergence.

Schemes effectively handle non-linear fluid-structure interactions.

Abstract

We consider a non-linear extension of Biot's model for poromechanics, wherein both the fluid flow and mechanical deformation are allowed to be non-linear. We perform an implicit discretization in time (backward Euler) and propose two iterative schemes for solving the non-linear problems appearing within each time step: a splitting algorithm extending the undrained split and fixed stress methods to non-linear problems, and a monolithic L-scheme. The convergence of both schemes is shown rigorously. Illustrative numerical examples are presented to confirm the applicability of the schemes and validate the theoretical results.