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.

翻译：多孔弹性问题的Biot模型通过Signorini接触条件进行了推广。由此产生的Biot接触问题被表述并分析为一个具有摄动鞍点结构的两场变分不等式问题。我们提出了一般情形以及$hp$-FE离散化的先验误差分析，包括收敛性及后者的保证收敛速率。此外，我们推导了一族可靠的且高效的基于残差的后验误差估计子，并阐述了如何应用一种简单高效的原-对偶积极集求解器来求解离散Galerkin问题。数值结果验证了我们的理论发现，并表明对于二维问题，通过自适应方案可以实现最优的收敛速率，特别是指数收敛速率。