  In this work, we present a new stabilization method aimed at removing
spurious oscillations in the pressure approximation of Biot's model for
poroelasticity with low permeabilities and/or small time steps. We consider
different finite-element discretizations and illustrate how not only does such
a stabilized scheme provide numerical solutions that are free of non-physical
oscillations, but it also allows one to iterate the fluid and mechanics
problems in a fashion similar to the well-known fixed-stress split method. The
resulting solution method is convergent without the necessity for additional
terms to stabilize the iteration. Finally, we present numerical results
illustrating the robust behavior of both the stabilization and iterative solver
with respect to the physical and discretization parameters of the model.
