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：针对线性椭圆问题的内部惩罚间断伽辽金(dG)方法，我们提出了一种新的残差型能量范数后验误差分析。新的误差界同样适用于由具有广泛多边形/多面体形状的单元构成的网格。单纯形和/或箱型单元的情况作为特例被包含在此分析中。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，本分析推广了已知的dG方法后验误差界，特别允许每个单元上存在任意数量的非规则悬挂节点。证明过程依赖于一种新的协调恢复策略结合亥姆霍兹分解公式。所得的后验误差界包含沿单元面切向导数的跳跃项。对于若干实际情形，也证明了局部下界。数值实验进一步揭示了所推导的后验误差界作为误差估计器的实用价值。