This paper focuses on the superconvergence analysis of the Hessian recovery technique for the $C^0$ Interior Penalty Method (C0IP) in solving the biharmonic equation. We establish interior error estimates for C0IP method that serve as the superconvergent analysis tool. Using the argument of superconvergence by difference quotient, we prove superconvergent results of the recovered Hessian matrix on translation-invariant meshes. The Hessian recovery technique enables us to construct an asymptotically exact ${\it a\, posteriori}$ error estimator for the C0IP method. Numerical experiments are provided to support our theoretical results.

翻译：本文聚焦于求解双调和方程时$C^0$内部罚方法（C0IP）中Hessian矩阵恢复技术的超收敛性分析。我们建立了C0IP方法的内部误差估计，将其作为超收敛分析工具。利用差商超收敛论证，在平移不变网格上证明了恢复Hessian矩阵的超收敛结果。Hessian矩阵恢复技术使我们能够为C0IP方法构造渐近精确的${\it 后验}$误差估计器。数值实验验证了理论结果。