Analysis of a multiphysics finite element method for a poroelasticity model

This article concerns finite element approximations of a quasi-static poroelasticity model in displacement-pressure formulation, which describes the dynamics of poro-elastic materials under an applied mechanical force on the boundary. To better describe the multiphysics process of deformation and diffusion for poroelastic materials, we first present a reformulation of the original model by introducing two pseudo-pressures. We then propose a time-stepping algorithm that decouples the reformulated partial differential equation (PDE) problem at each time step into two sub-problems: one of which is a generalized Stokes problem for the displacement vector field (of the solid network of the poroelastic material) along with one pseudopressure field, and the other is a diffusion problem for the other pseudo-pressure field (of the solvent of the material). To make this multiphysics approach feasible numerically, two critical issues must be resolved: the first one is the uniqueness of the generalized Stokes problem and the other is to find a good boundary condition for the diffusion equation, so that it also becomes uniquely solvable. To address the first issue, we discover certain conserved quantities for the PDE solution that provide ideal candidates for a needed boundary condition for the pseudo-pressure field. The solution to the second issue is to use the generalized Stokes problem to generate a boundary condition for the diffusion problem. A practical advantage of the time-stepping algorithm allows one to use any convergent Stokes solver (and its code) together with any convergent diffusion equation solver (and its code) to solve the poroelasticity model. In this article, the Taylor-Hood mixed finite element method combined with the P-1-conforming finite element method is used as an example to demonstrate the viability of the proposed multiphysics approach. It is proved that the solutions of the fully discrete finite element methods fulfill a discrete energy law, which mimics the differential energy law satisfied by the PDE solution and converges optimally in the energy norm. Moreover, it is showed that the proposed formulation also has a built-in mechanism to overcome so-called 'locking phenomenon' associated with the numerical approximations of the poroelasticity model. Numerical experiments are presented to show the performance of the proposed approach and methods, and to demonstrate the absence of 'locking phenomenon' in our numerical experiments. This article also presents a detailed PDE analysis for the poroelasticity model, especially it is proved that this model converges to the well-known Biot's consolidation model from soil mechanics as the constrained specific storage coefficient tends to zero. As a result, the proposed approach and methods are robust under such a limit process.