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

We investigate the behavior of the approximate solution of Biot's con-solidation 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 appro-priately, 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.

Automated summary

The behavior of the approximate solution of Biot's consolidation model is investigated using a weighted least-squares finite element method. The model describes fluid flow in a deformable porous medium and includes variables for fluid pressure, velocity, and displacement. The weighted least-squares functional is defined based on the stress-displacement formulation, with a symmetry condition for the stress and a weight dependent on the time step size. The method is validated through a priori error estimation and numerical examples.