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

Yanan He, Zhihao Ge

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 $κ_1$ and $κ_3$ remain bounded and non-zero even as $λ$ 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.

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

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 and remain bounded and non-zero even as 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 , we give the stability analysis and derive optimal error estimates for the displacement vector and the pressure in both -norm and -norm. In particular, the -norm error estimate for the displacement , which was not present in previous literature, is established here through an auxiliary problem and a Poincar 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.
Paper Structure (11 sections, 14 theorems, 147 equations, 6 figures, 6 tables, 1 algorithm)

This paper contains 11 sections, 14 theorems, 147 equations, 6 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Multiphysics reformulation and PDE analysis
  3. Multiphysics reformulation
  4. PDE analysis
  5. Fully discrete multiphysics finite element method
  6. Multiphysics finite element method
  7. Stability analysis
  8. Error estimates
  9. $L^{2}$ error estimation of displacement $\mathbf {u}$
  10. Numerical tests
  11. Conclusion

Key Result

Lemma 2.2

\newlabellemma2.3 The function $a_{ij}(\cdot,\varepsilon(\mathbf{u}))$ is measurable in $\Omega$ for all $\varepsilon(\mathbf{u})\in \mathbb{R}^{2\times2}$, and $a_{ij}(x,\cdot)$ is continuous in $\mathbb{R}^{2\times2}$ for almost all $x\in \Omega$. Also there exists $C>0$ such that for all $\varepsilon(\mathbf{u})\in \mathbb{R}^{2\times2}$ and for almost all $x\in \Omega$.

Figures (6)

  • Figure 4.1: Surface plot of the computed displacement $u_1$ at the terminal time $T$ .
  • Figure 4.2: Surface plot of the computed displacement $u_2$ at the terminal time $T$.
  • Figure 4.3: Surface plots of the computed pressure $p$ and exact pressure $p$ at the terminal time $T$.
  • Figure 4.4: Surface plot of the computed displacement $u_1$ at the terminal time $T$ .
  • Figure 4.5: Surface plot of the computed displacement $u_2$ at the terminal time $T$.
  • ...and 1 more figures

Theorems & Definitions (27)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Remark 2.1
  • Lemma 2.7
  • ...and 17 more