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：Signorini接触条件扩大了生物多孔性问题的范围,由此产生的生物接触问题被拟订为并分析成一个不连续的马鞍结构的两个实地变异性不平等问题。我们为一般和美元-FE的分解提供了先验错误分析,包括后者的趋同率和保证汇合率。此外,我们从一个可靠和高效的残余物中产生一个后继误差估计器,并阐述如何应用一个简单和高效的原始多功能成套解决器来解决离散的Galerkin问题。 数字结果强调了我们的理论结论,并表明两个维问题的适应计划可以实现最佳的、特别是指数性汇合率。