Paper
Optimal Error Estimates of a new Multiphysic Finite Element Method for Nonlinear Poroelasticity model with Hencky-Mises Stress Tensor
Authors
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.
Metadata
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Raw Data (Debug)
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2602.19457v1</id>\n <title>Optimal Error Estimates of a new Multiphysic Finite Element Method for Nonlinear Poroelasticity model with Hencky-Mises Stress Tensor</title>\n <updated>2026-02-23T02:58:23Z</updated>\n <link href='https://arxiv.org/abs/2602.19457v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2602.19457v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>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.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='math.NA'/>\n <category scheme='http://arxiv.org/schemas/atom' term='math.AP'/>\n <published>2026-02-23T02:58:23Z</published>\n <arxiv:primary_category term='math.NA'/>\n <author>\n <name>Yanan He</name>\n </author>\n <author>\n <name>Zhihao Ge</name>\n </author>\n </entry>"
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