Error analysis for the approximation of a flow in deformable porous media with nonlinear strain-stress relation

TL;DR

This paper develops and analyzes a numerical scheme for simulating fluid flow in deformable porous media with nonlinear elastic stress-strain behavior, ensuring convergence and demonstrating effectiveness through experiments.

Contribution

The paper introduces a semi-implicit finite element scheme for nonlinear poroelastic flow, proving its stability and convergence under small nonlinear perturbations.

Findings

The scheme converges with a priori estimates under small nonlinear effects.

Numerical experiments confirm the scheme's efficiency and ability to capture nonlinear phenomena.

The model effectively describes slow flow in deformable porous media with nonlinear elasticity.

Abstract

We study a model describing the slow flow of a fluid through a deformable, porous, elastic solid undergoing small deformations. The stress-strain relationship of the solid incorporates nonlinear effects, formulated as a perturbation of the classical linear elasticity. To approximate the coupled system, we introduce a discrete scheme based on a first order semi-implicit time integration scheme combined with a standard finite element spatial discretization. We establish the existence and uniqueness of the discrete solution and derive a priori convergence estimates under the assumption that the nonlinear perturbations remain sufficiently small. Finally, we demonstrate the efficiency of the proposed scheme through numerical experiments that also highlight the nonlinear phenomena captured by the model.