  A stabilized conforming mixed finite element method for the three-field
(displacement, fluid flux and pressure) poroelasticity problem is developed and
analyzed. We use the lowest possible approximation order, namely piecewise
constant approximation for the pressure and piecewise linear continuous
elements for the displacements and fluid flux. By applying a local pressure
jump stabilization term to the mass conservation equation we ensure stability
and avoid pressure oscillations. Importantly, the discretization leads to a
symmetric linear system. For the fully discretized problem we prove existence
and uniqueness, an energy estimate and an optimal a-priori error estimate,
including an error estimate for the divergence of the fluid flux. Numerical
experiments in 2D and 3D illustrate the convergence of the method, show the
effectiveness of the method to overcome spurious pressure oscillations, and
evaluate the added mass effect of the stabilization term.
