Optimal Error Estimates of a new Multiphysic Finite Element Method for Nonlinear Poroelasticity model with Hencky-Mises Stress Tensor

TL;DR

This paper introduces a new multiphysics finite element method for nonlinear poroelasticity with Hencky-Mises stress, providing optimal error estimates and demonstrating stability and convergence through numerical tests.

Contribution

The paper develops a stable, fully discrete multiphysics finite element method with new error estimates, including the first $L^2$-norm error for displacement in this context.

Findings

Establishment of existence and uniqueness of the weak solution.

Derivation of optimal error estimates for displacement and pressure.

Numerical verification confirming theoretical convergence rates.

Abstract

In this paper, we develop a new multiphysics finite element method for a nonlinear poroelastic model with Hencky-Mises stress tensor. By introducing some new notations, we reformulate the original model into a fluid-fluid coupling problem, which is viewed as a generalized nonlinear Stokes sub-problem combined with a reaction-diffusion sub-problem. Then, we establish the existence and uniqueness of the weak solution for the reformulated problem, and propose a stable, fully discrete multiphysics finite element method which employs Lagrangian finite element pairs for spatial discretization and a backward Euler scheme for temporal discretization. By ensuring the parameters $\kappa_1$ and $\kappa_3$ remain bounded and non-zero even as $\lambda$ tends to infinity, the proposed method maintains stability for a wide range of Lagrangian element pairs. Based on the continuity and monotonicity of the nonlinear term $\mathcal{N}(\varepsilon(\mathbf{u}_h^{n}))$, we give the stability analysis and derive optimal error estimates for the displacement vector $\mathbf{u}$ and the pressure $p$ in both $L^2$-norm and $H^1$-norm. In particular, the $L^2$-norm error estimate for the displacement $\mathbf{u}$, which was not present in previous literature, is established here through an auxiliary problem and a Poincar$\acute{e}$ inequality. Also, we present numerical tests to verify the theoretical analysis, and the results confirm the optimal convergence rates. Finally, we draw conclusions to summarize the work.