Performance of a Mixed Least‐Squares Finite Element Formulation With Application in Poroelasticity

Abstract

A mixed least‐squares finite element method (LSFEM), previously proposed for the theory of porous media (TPM), is further investigated in this work. Finite deformation of a fully saturated, incompressible solid–fluid binary system is studied using the LSFEM. The stress and deformation of the solid phase, along with the pore pressure and velocity of the liquid phase, constitute four independent fields in this formulation. The normal‐flux preserving, vector‐valued Raviart–Thomas (RT) functions are employed to discretize the stress and velocity fields, while the displacement and pressure fields are approximated using conventional Lagrange polynomials ( P ). The LSFEM avoids the intricacies associated with satisfying the inf‐sup (or LBB) condition required in mixed Galerkin finite element methods to ensure stability. Another key advantage of the LSFEM is that non‐self‐adjoint differential operators also result in a symmetric positive‐definite system of matrices. The error minimization approach adapted in LSFEM makes the role of residual weights crucial in deciding the convergence characteristics. Therefore, in this study, the performance of higher‐order RT‐P elements is evaluated against the canonical consolidation problem. The suitability of the developed LSFEM for the simulation of fluid‐saturated porous media with incompressible constituents is thus demonstrated.