Contact problems in porous media

TL;DR

This paper extends the Biot poroelasticity model by incorporating contact conditions, providing a comprehensive analysis of the resulting variational inequality, error estimates, and efficient numerical solution methods with adaptive schemes.

Contribution

It introduces a novel formulation of the Biot contact problem as a variational inequality, along with error analysis, residual estimators, and an active set solver for improved numerical solutions.

Findings

Optimal convergence rates achieved with adaptive schemes

Guaranteed convergence for the discretization methods

Numerical results confirm theoretical error estimates

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 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.