SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1273-1298, January 2021.
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 generalized Helmholtz decomposition of the error. This is the first $hp$-version error estimator for the biharmonic problem in two and three dimensions. The practical behavior of the estimator is investigated through numerical examples in two and three dimensions.

中文翻译：

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基于残基的双调和问题的$ hp $-不连续Galerkin离散后验误差估计

SIAM数值分析学报，第59卷，第3期，第1273-1298页，2021年1月。
我们针对二维和三维双谐波问题引入了一种基于残差的后验误差估计器，该误差估计器用于新颖的$ hp $版本内部罚分不连续Galerkin方法。我们证明误差估计为误差提供了一个上限和一个局部下限，并且该下限对局部网格大小具有鲁棒性，但对局部多项式度却不具有鲁棒性。就多项式而言，次优是完全明确的，并且至多以代数形式增长。我们的分析不需要存在符合$ \ mathcal {C} ^ 1 $的分段多项式空间，而是基于对$ H ^ 2 $空间的离散解的椭圆重构以及该函数的广义Helmholtz分解。错误。这是二维和双维双谐波问题的第一个$ hp $版本误差估计器。