A time-nonlocal multiphysics finite element method with Crank-Nicolson scheme for poroelasticity model with secondary consolidation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910112344440832 |
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| author | Ge, Zhihao He, Yanan |
| author_facet | Ge, Zhihao He, Yanan |
| contents | The paper studies a time-nonlocal multiphysics finite element method with Crank-Nicolson scheme for poroelasticity model with secondary consolidation. For the case where the physical parameters $λ,λ^*$ and $c_0$ are all finite positive constants, by introducing two auxiliary variables-the fluid content $η$ and the generalized pressure $ξ$ -- the original strongly coupled poroelasticity model is reformulated into a generalized Stokes equation with time integral terms and a diffusion equation. The reformulated model not only reveals the underlying multiphysics processes in the original model, but also exhibits time-nonlocal characteristics. A time-nonlocal multiphysics finite element method is designed for the reformulated model: the spatial discretization employs high order Taylor-Hood mixed finite element method, and the temporal discretization adopts the Crank-Nicolson scheme. The time integral terms are approximated using the composite trapezoidal rule, and the integral terms $J_ξ^n$ and $J_η^n$ are introduced for real-time updates, which not only avoids repeated calculations and improves efficiency, but also maintains second-order temporal accuracy. The existence and uniqueness of weak solutions for the reformulated model are proved via energy estimate methods, the stability of the fully discrete time-nonlocal multiphysics finite element method is established, and optimal-order error estimates are derived using projection operator techniques. Finally, numerical example verified the theoretical results and compared the long-time convergence of the Crank-Nicolson scheme and the backward Euler scheme. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06815 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A time-nonlocal multiphysics finite element method with Crank-Nicolson scheme for poroelasticity model with secondary consolidation Ge, Zhihao He, Yanan Numerical Analysis The paper studies a time-nonlocal multiphysics finite element method with Crank-Nicolson scheme for poroelasticity model with secondary consolidation. For the case where the physical parameters $λ,λ^*$ and $c_0$ are all finite positive constants, by introducing two auxiliary variables-the fluid content $η$ and the generalized pressure $ξ$ -- the original strongly coupled poroelasticity model is reformulated into a generalized Stokes equation with time integral terms and a diffusion equation. The reformulated model not only reveals the underlying multiphysics processes in the original model, but also exhibits time-nonlocal characteristics. A time-nonlocal multiphysics finite element method is designed for the reformulated model: the spatial discretization employs high order Taylor-Hood mixed finite element method, and the temporal discretization adopts the Crank-Nicolson scheme. The time integral terms are approximated using the composite trapezoidal rule, and the integral terms $J_ξ^n$ and $J_η^n$ are introduced for real-time updates, which not only avoids repeated calculations and improves efficiency, but also maintains second-order temporal accuracy. The existence and uniqueness of weak solutions for the reformulated model are proved via energy estimate methods, the stability of the fully discrete time-nonlocal multiphysics finite element method is established, and optimal-order error estimates are derived using projection operator techniques. Finally, numerical example verified the theoretical results and compared the long-time convergence of the Crank-Nicolson scheme and the backward Euler scheme. |
| title | A time-nonlocal multiphysics finite element method with Crank-Nicolson scheme for poroelasticity model with secondary consolidation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2604.06815 |