We develop the \textit{a posteriori} error analysis of three mixed finite element formulations for rotation-based equations in elasticity, poroelasticity, and interfacial elasticity-poroelasticity. The discretisations use $H^1$-conforming finite elements of degree $k+1$ for displacement and fluid pressure, and discontinuous piecewise polynomials of degree $k$ for rotation vector, total pressure, and elastic pressure. Residual-based estimators are constructed, and upper and lower bounds (up to data oscillations) for all global estimators are rigorously derived. The methods are all robust with respect to the model parameters (in particular, the Lam\'e constants), they are valid in 2D and 3D, and also for arbitrary polynomial degree $k\geq 0$. The error behaviour predicted by the theoretical analysis is then demonstrated numerically on a set of computational examples including different geometries on which we perform adaptive mesh refinement guided by the \textit{a posteriori} error estimators.

中文翻译：

---

基于旋转的弹性/多孔弹性耦合公式的稳健后验误差分析

我们为弹性、多孔弹性和界面弹性-多孔弹性中基于旋转的方程开发了三种混合有限元公式的\textit{a后验}误差分析。离散化使用 $H^1$-符合 $k+1$ 次的有限元来表示位移和流体压力，使用次数为 $k$ 的不连续分段多项式来表示旋转矢量、总压力和弹性压力。构建基于残差的估计量，并严格推导出所有全局估计量的上限和下限（直至数据振荡）。这些方法对于模型参数（特别是 Lam\'e 常数）都是稳健的，它们在 2D 和 3D 中有效，也适用于任意多项式次数 $k\geq 0$。