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

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.