We introduce a residual-based a posteriori error estimator for a novel $hp$-version interior penalty discontinuous Galerkin method for the biharmonic problem in two and three dimensions. We prove that the error estimate provides an upper bound and a local lower bound on the error, and that the lower bound is robust to the local mesh size but not the local polynomial degree. The suboptimality in terms of the polynomial degree is fully explicit and grows at most algebraically. Our analysis does not require the existence of a $\mathcal{C}^1$-conforming piecewise polynomial space and is instead based on an elliptic reconstruction of the discrete solution to the $H^2$ space and a generalised Helmholtz decomposition of the error. This is the first $hp$-version error estimator for the biharmonic problem in two and three dimensions. The practical behaviour of the estimator is investigated through numerical examples in two and three dimensions.

翻译：我们引入了一种基于残余的后置误差估计器, 用于对两维和三个维度的双声调问题进行新颖的 $hp$- version 内置惩罚不连续的 Galerkin 方法。 我们证明错误估计提供了上界和局部下界的误差, 下界对本地网目大小是强健的, 但对本地多面度不是。 多面度的亚优化度是完全清晰的, 并在最远的代数上增长。 我们的分析并不要求存在一个 $\mathcal{C ⁇ 1$- 匹配的片状多面空间, 而是基于对$H$2$空间的离散解决方案进行椭圆式重建, 以及错误的普通化 Helmholtz 分解。 这是两个维度和三个维度的首个$hp$- version错误估计器。 我们的分析并不需要存在一个 $\ mathcal{C ⁇ 1$- $- commonnomial space space space space, 而是基于两个和三个维的数值实例来调查估算器的实际行为。