  In this work, we consider the popular P1-RT0-P0 discretization of the
three-field formulation of Biot's consolidation problem. Since this
finite-element formulation does not satisfy an inf-sup condition uniformly with
respect to the physical parameters, several issues arise in numerical
simulations. For example, when the permeability is small with respect to the
mesh size, volumetric locking may occur. Thus, we propose a stabilization
technique that enriches the piecewise linear finite-element space of the
displacement with the span of edge/face bubble functions. We show that for
Biot's model this does give rise to discretizations that are uniformly stable
with respect to the physical parameters. We also propose a perturbation of the
bilinear form, which allows for local elimination of the bubble functions and
provides a uniformly stable scheme with the same number of degrees of freedom
as the classical P1-RT0-P0 approach. We prove optimal stability and error
estimates for this discretization. Finally, we show that this scheme can also
be successfully applied to Stokes' equations, yielding a discrete problem with
optimal approximation properties and with minimum number of degrees of freedom
(equivalent to a P1-P0 discretization). Numerical tests confirm the theory for
both poroelastic and Stokes' test problems.
