A mixed least‐squares finite element formulation within the framework of the theory of porous media

In the present contribution a mixed least‐squares finite element method (LSFEM) based on the Theory of Porous Media (TPM) is presented. In detail, we investigate an incompressible binary model consisting of the phases solid and liquid. The main idea is based on the modeling of saturated porous structures. The resulting finite element is a four‐field formulation in terms of solid displacements, liquid pressure, mixture stresses and a new variable related to the pressure gradient. The conforming discretization of the unknowns in the spaces H(div) and H1 is realized by vector‐valued Raviart‐Thomas and standard Lagrange functions. Finally, a numerical example for liquid saturated porous structures considering an incompressible, linear elastic material behavior at small deformations, demonstrates the applicability of the LSFEM approach to the TPM.