论文标题
基于残差的后验错误估计$ \ mathbf {hp} $ - 不连续的galerkin biharmonic问题离散
Residual-based a posteriori error estimates for $\mathbf{hp}$-discontinuous Galerkin discretisations of the biharmonic problem
论文作者
论文摘要
我们引入了一种基于残差的后验误差估计器，用于新型$ hp $ version内部罚款不连续的盖尔金方法，用于在两个维度和三个维度中用于双旋转问题。我们证明，误差估计值在误差上提供了上限和局部下限，并且下限对局部网格大小具有鲁棒性，而不是局部多项式程度。从多项式程度上的次要性是完全显式的，最多可以在代数上生长。我们的分析不需要存在$ \ MATHCAL {C}^1 $ - 合并分段多项式空间，而是基于离散解决方案的椭圆形重构到$ h^2 $空间和通用的Helmholtz分解错误。这是两个维度和三个维度的Biharmonic问题的第一个$ HP $ version误差估计器。估计量的实际行为通过两个维度和三个维度的数值示例进行了研究。
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.
