Overview
A time-nonlocal multiphysics finite element method, employing a Crank-Nicolson scheme, has been developed for a poroelasticity model that includes secondary consolidation. This approach involves reformulating the original strongly coupled poroelasticity model into a generalized Stokes equation characterized by time integral terms and a diffusion equation.
Research Context
The research addresses poroelasticity models, specifically those incorporating secondary consolidation. In scenarios where physical parameters $\lambda$, $\lambda^*$, and $c_0$ are finite positive constants, the inherent complexities of the original strongly coupled poroelasticity model are acknowledged. The reformulation aims to clarify the underlying multiphysics processes and highlight time-nonlocal characteristics.
Approach
The methodology initiated by introducing two auxiliary variables: the fluid content $\eta$ and the generalized pressure $\xi$. This allowed for the reformulation of the original poroelasticity model into a generalized Stokes equation that includes time integral terms, and a diffusion equation. This reformulated model not only reveals the intrinsic multiphysics processes but also exhibits time-nonlocal properties.
The designed time-nonlocal multiphysics finite element method uses a high-order Taylor-Hood mixed finite element method for spatial discretization. Temporal discretization is performed using the Crank-Nicolson scheme. The time integral terms are approximated via the composite trapezoidal rule. To enhance efficiency and maintain desired accuracy, integral terms $J_{\xi}^n$ and $J_{\eta}^n$ are introduced for real-time updates, which avoids repetitive calculations. This strategy also ensures second-order temporal accuracy.
Findings
- The original strongly coupled poroelasticity model, when physical parameters $\lambda,\lambda^*$ and $c_0$ are finite positive constants, can be reformulated into a generalized Stokes equation with time integral terms and a diffusion equation through the introduction of auxiliary variables $\eta$ (fluid content) and $\xi$ (generalized pressure).
- The reformulated model elucidates the underlying multiphysics processes of the original model and demonstrates time-nonlocal characteristics.
- The time-nonlocal multiphysics finite element method, utilizing high-order Taylor-Hood mixed finite elements for spatial discretization and the Crank-Nicolson scheme for temporal discretization, maintains second-order temporal accuracy.
- The approximation of time integral terms via the composite trapezoidal rule, in conjunction with the introduction of $J_{\xi}^n$ and $J_{\eta}^n$ for real-time updates, improved computational efficiency by avoiding repeated calculations.
- Existence and uniqueness of weak solutions for the reformulated model were proven using energy estimate methods.
- The stability of the fully discrete time-nonlocal multiphysics finite element method was established.
- Optimal-order error estimates were derived through the application of projection operator techniques.
- Numerical examples verified the theoretical results and facilitated a comparison of the long-time convergence between the Crank-Nicolson scheme and the backward Euler scheme.