A time-nonlocal multiphysics finite element method with Crank-Nicolson scheme for poroelasticity model with secondary consolidation

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 $\lambda,\lambda^*$ and $c_0$ are all finite positive constants, by introducing two auxiliary variables-the fluid content $\eta$ and the generalized pressure $\xi$ -- 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 $J_{\xi}^n$ and $J_{\eta}^n$ 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.