Contact problems in porous media

The Biot problem of poroelasticity is extended by Signorini contact conditions. The resulting Biot contact problem is formulated and analyzed as a two field variational inequality problem of a perturbed saddle point structure. We present an a priori error analysis for a general as well as for a $hp$-FE discretization including convergence and guaranteed convergence rates for the latter. Moreover, we derive a family of reliable and efficient residual based a posteriori error estimators, and elaborate how a simple and efficient primal-dual active set solver can be applied to solve the discrete Galerkin problem. Numerical results underline our theoretical finding and show that optimal, in particular exponential, convergence rates can be achieved by adaptive schemes for two dimensional problems.