Contact Problems in Porous Media

Abstract 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 an hp -FE discretization including convergence and guaranteed convergence rates for the latter. In particular, these rates are always optimal in the discretization parameters for the fluid pressure, and can be linear for the lowest order h -version. Moreover, we state a residual based a posteriori error estimator. Numerical results underline our theoretical findings and show that optimal, in particular exponential, convergence rates can be achieved by adaptive schemes for two-dimensional problems.