  This paper concerns with 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 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). In the paper, 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.
