In recent years, a lot of effort has been dedicated to the theoretical and numerical treatment of models for fluid flow and deformation in porous media. Our model will be based on the classical phenomenological approach by Biot [2] using the concept of effective stresses. Modern formulations based on multiphase mixture theories were developed in the last 30 years, see the monograph by de Boer [12] for an overview. A lot of current research activity concerned with the numerical treatment of such coupled problems is devoted to the extension to two-phase flow and elasto-plastic deformation models.

The classical Biot consolidation problem assumes a fully saturated porous medium and a linearly elastic material law leading to a linear parabolic system. We restrict our attention to this linear problem since our aim is to analyze a least-squares mixed finite element method in the simplest possible situation. The novelty of our least-squares approach is the introduction of approximation spaces for the fluid flux and the stress tensor in addition to the primary variables fluid pressure and displacement field.