Stabilized Lowest-Order Finite Element Approximation for Linear Three-Field Poroelasticity

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 two and three dimensions illustrate the convergence of the method, show its effectiveness in overcoming spurious pressure oscillations, and evaluate the added mass effect of the stabilization term.