We present a finite element discretisation to model the interaction between a poroelastic structure and an elastic medium. The consolidation problem considers fully coupled deformations across an interface, ensuring continuity of displacement and total traction, as well as no-flux for the fluid phase. Our formulation of the poroelasticity equations incorporates displacement, fluid pressure, and total pressure, while the elasticity equations adopt a displacement-pressure formulation. Notably, the transmission conditions at the interface are enforced without the need for Lagrange multipliers. We demonstrate the stability and convergence of the divergence-conforming finite element method across various polynomial degrees. The a priori error bounds remain robust, even when considering large variations in intricate model parameters such as Lam\'e constants, permeability, and storativity coefficient. To enhance computational efficiency and reliability, we develop residual-based a posteriori error estimators that are independent of the aforementioned coefficients. Additionally, we devise parameter-robust and optimal block diagonal preconditioners. Through numerical examples, including adaptive scenarios, we illustrate the scheme's properties such as convergence and parameter robustness.

翻译：本文提出一种有限元离散方法，用于模拟多孔弹性结构与弹性介质之间的相互作用。固结问题考虑了界面上的完全耦合变形，确保位移与总牵引力的连续性，同时流体相无通量。多孔弹性方程采用位移、流体压力及总压力的联合公式，而弹性方程则采用位移-压力公式。值得注意的是，界面处的传递条件无需引入拉格朗日乘子即可强制执行。我们证明了该散度相容有限元方法在不同多项式阶数下的稳定性与收敛性。先验误差界对于拉梅常数、渗透率及储水系数等复杂模型参数的大幅变化仍保持稳健。为提升计算效率与可靠性，我们发展了与前述系数无关的残差型后验误差估计器。此外，还设计了参数鲁棒且最优的分块对角预处理子。通过数值算例（包括自适应场景），我们展示了该方案的收敛性与参数鲁棒性等特性。