We present a new residual-type energy-norm a posteriori error analysis for interior penalty discontinuous Galerkin (dG) methods for linear elliptic problems. The new error bounds are also applicable to dG methods on meshes consisting of elements with very general polygonal/polyhedral shapes. The case of simplicial and/or box-type elements is included in the analysis as a special case. In particular, for the upper bounds, an arbitrary number of very small faces are allowed on each polygonal/polyhedral element, as long as certain mild shape regularity assumptions are satisfied. As a corollary, the present analysis generalizes known a posteriori error bounds for dG methods, allowing in particular for meshes with an arbitrary number of irregular hanging nodes per element. The proof hinges on a new conforming recovery strategy in conjunction with a Helmholtz decomposition formula. The resulting a posteriori error bound involves jumps on the tangential derivatives along elemental faces. Local lower bounds are also proven for a number of practical cases. Numerical experiments are also presented, highlighting the practical value of the derived a posteriori error bounds as error estimators.

翻译：本文针对线性椭圆问题的内惩罚间断伽辽金（dG）方法，提出了一种新型残差型能量范数后验误差分析。该误差界同样适用于由具有一般多边形/多面体形状单元构成的网格。单纯形和/或盒形单元的情况作为特例包含在此分析中。特别地，对于上界，只要满足某些温和的形状正则性假设，每个多边形/多面体单元上允许存在任意数量的极小面。作为推论，本分析推广了已有的dG方法后验误差界，特别允许每个单元上存在任意数量的不规则悬挂节点。证明依赖于一种新的协调恢复策略与亥姆霍兹分解公式的结合。所得后验误差界涉及沿单元面的切向导数跳跃项。还针对若干实际情形证明了局部下界。给出了数值实验，突出显示了导出的后验误差界作为误差估计器的实用价值。