MULTIPHYSICS FINITE ELEMENT METHODS FOR A POROELASTICITY MODEL∗

Abstract. This paper concerns with finite element approximations of a quasi-static poroe-lasticity 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 poro-elastic materials, we first present a reformulation of the original model by introducing two pseudo-pressures, one of them is shown to satisfy a diffusion equation, we then propose a time-stepping algorithm which decouples (or couples) the reformulated 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 poro-elastic material) along with one pseudo-pressure 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 issue 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 which 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 con