A Parallel Solver with Multiphysics Finite Element Method for Poroelasticity Coupled with Elasticity Model

In this paper, we propose a parallel solver for solving the quasi-static linear poroelasticity coupled with linear elasticity model in the Lagrange multiplier framework. Firstly, we reformulate the model into a coupling of the nearly incompressible elasticity and an unsteady affection-diffusion equations by setting new variable ``elastic pressure" and ``volumetric fluid content". And we introduce a Lagrange multiplier to guarantee the normal stress continuity on the interface. Then, we give the variational formulations in each subdomain and choose the $\boldsymbol{P}_k$-$P_1$-$P_1$ mixed finite element tuple for poroelasticity subdomain, and $\boldsymbol{P}_k$-$P_1$ finite element pair ($k=1,2$) for elasticity subdomain and the backward Euler scheme for time. Also, we propose a parallel solver for solving the fully discrete scheme at each time step-- the FETI method with a classical FETI preconditioner for solving the Lagrange multiplier and calculating the subproblems in each subdomain in parallel. And we show several numerical tests to validate the computational efficiency and the convergence error order, and we consider Barry-Mercer's model as the benchmark test to show that there no oscillation in the computed pressure. Finally, we draw conclusions to summarize the main results of this paper.