Optimal error estimates of multiphysics finite element method for a nonlinear poroelasticity model with nonlinear stress-strain relations

TL;DR

This paper develops and analyzes a finite element method for a nonlinear poroelasticity model, providing error estimates and stability analysis, with numerical tests confirming the theoretical results.

Contribution

It introduces a new reformulation of the nonlinear poroelasticity model and provides the first comprehensive error estimates for its finite element discretization.

Findings

The proposed method is stable and convergent.

Error estimates are established for the fully discrete scheme.

Numerical tests confirm the theoretical error bounds.

Abstract

In this paper, we study the numerical algorithm for a nonlinear poroelasticity model with nonlinear stress-strain relations. By using variable substitution, the original problem can be reformulated to a new coupled fluid-fluid system, that is, a generalized nonlinear Stokes problem of displacement vector field related to pseudo pressure and a diffusion problem of other pseudo pressure fields. A new technique is used to get the existence and uniqueness of the solution of the reformulated model and a fully discrete nonlinear finite element method is proposed to solve the model numerically. The multiphysics finite element is used to get the discretization of the space variable and the backward Euler method is taken as the time-stepping method in the fully discrete case. Stability analysis and the error estimation are given for the fully discrete case and numerical test are taken to verify the theoretical results.