A least-squares mixed finite element method for Biot's consolidation problem in porous media

A least-squares mixed finite element method for the coupled problem of flow and deformation is presented and analyzed in this paper. For the analysis, we restrict ourselves to fully saturated conditions for the flow process and to a linearly elastic material law for the deformation process. This is known in the literature as Biot's consolidation problem. For simplicity, the analysis is presented for the problem in two space dimensions. Our least-squares approach is motivated by the fact that all process variables, i.e., fluid pressure and flux as well as displacement field and stress tensor, are approximated directly by suitable finite element spaces. Ellipticity of the corresponding variational formulation is proven for the stationary case as well as for the subproblems arising at each step of an implicit time discretization in the general time-dependent case. Standard H-1-conforming piecewise linear and quadratic finite elements are used for the fluid pressure and for (each component of) the displacement, respectively. For the flux and stress components, the H(div)-conforming Raviart-Thomas spaces (of lowest order) are used. Computational results are presented for some two-dimensional test problems.