Equal Lower-order Finite Elements of Least-squares Type in Biot Poroelasticity Modeling

We investigate the behavior of the approximate solution of Biot's consolidation model using a weighted least-squares (WLS) finite element method. The model describes the fluid flow in a deformable porous medium, with variables for fluid pressure, velocity, and displacement. The WLS functional is defined based on the stress-displacement formulation, with the symmetry condition of the stress and the weight that depends on the time step size for the temporal discretization of the model. An a priori error estimate for the first-order linearized least squares (LS) system is analyzed, and its validity is confirmed through numerical results. By using continuous piecewise linear finite element spaces for all variables and adjusting the weight appropriately, we obtain optimal error convergence rates for all variables. Additionally, we present two numerical examples to demonstrate the implementation of the WLS method for benchmark problems.