Error analysis for the approximation of a flow in deformable porous media with nonlinear strain-stress relation
Provides rigorous error analysis for a nonlinear poroelasticity model, relevant for geomechanics and biomedical applications.
The paper develops and analyzes a numerical scheme for fluid flow in deformable porous media with nonlinear stress-strain, proving convergence estimates and demonstrating efficiency via numerical experiments.
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.