In this paper, we study two residual-based a posteriori error estimators for the $C^0$ interior penalty method in solving the biharmonic equation in a polygonal domain under a concentrated load. The first estimator is derived directly from the model equation without any post-processing technique. We rigorously prove the efficiency and reliability of the estimator by constructing bubble functions. Additionally, we extend this type of estimator to general fourth-order elliptic equations with various boundary conditions. The second estimator is based on projecting the Dirac delta function onto the discrete finite element space, allowing the application of a standard estimator. Notably, we additionally incorporate the projection error into the standard estimator. The efficiency and reliability of the estimator are also verified through rigorous analysis. We validate the performance of these a posteriori estimates within an adaptive algorithm and demonstrate their robustness and expected accuracy through extensive numerical examples.

翻译：本文针对多边形域中承受集中载荷的双调和方程，研究了基于残差的两种后验误差估计器在$C^0$内部惩罚法中的应用。第一种估计器直接从模型方程推导得出，无需任何后处理技术。我们通过构造气泡函数严格证明了该估计器的有效性与可靠性。此外，我们将此类估计器推广至具有不同边界条件的一般四阶椭圆方程。第二种估计器基于将狄拉克δ函数投影到离散有限元空间，从而能够应用标准估计器。值得注意的是，我们还将投影误差纳入标准估计器中。通过严格的分析验证了该估计器的有效性与可靠性。我们在自适应算法中验证了这些后验估计的性能，并通过大量数值算例证明了其鲁棒性与预期精度。