Error Estimates of a Fully Discrete Multiphysics Finite Element Method for a Nonlinear Poroelasticity Model

TL;DR

This paper develops and analyzes a fully discrete multiphysics finite element method for a nonlinear poroelasticity model, providing error estimates and numerical validation without assumptions on the nonlinear stress-strain relation.

Contribution

It introduces a novel multiphysics reformulation and a fully discrete scheme with optimal error estimates for nonlinear poroelasticity, avoiding the locking phenomenon.

Findings

Optimal convergence order error estimates derived

Numerical examples confirm theoretical analysis

No locking phenomenon observed in simulations

Abstract

In this paper, we propose a multiphysics finite element method for a nonlinear poroelasticity model. To better describe the processes of deformation and diffusion, we firstly reformulate the nonlinear fluid-solid coupling problem into a fluid-fluid coupling problem by a multiphysics approach. Then we design a fully discrete time-stepping scheme to use multiphysics finite element method with $P_2-P_1-P_1$ element pairs for the space variables and backward Euler method for the time variable, and we adopt the Newton iterative method to deal with the nonlinear term. Also, we derive the discrete energy laws and the optimal convergence order error estimates without any assumption on the nonlinear stress-strain relation. Finally, we show some numerical examples to verify the rationality of theoretical analysis and there is no "locking phenomenon".