论文标题
多边形和多面部网格上不连续的Galerkin方法的后验误差估计值
A posteriori error estimates for discontinuous Galerkin methods on polygonal and polyhedral meshes
论文作者
论文摘要
我们提出了一种新的残留型能量 - 对于线性椭圆问题的内部罚款不连续的盖尔金（DG）方法的后验误差分析。新的误差界也适用于由具有非常通用的多边形/多面体形状的元素组成的网格的DG方法。分析中包括简单和/或盒子类型元素的情况。特别是，对于上边界，只要满足某些轻度形状的规律性假设，每个多边形/多面体元件就允许在每个多边形/多面体元件上进行任意数量的非常小的面孔。作为推论，目前的分析概括了DG方法的已知后验误差界限，尤其允许与任意数量的每个元素不规则悬挂节点的网格。证明与Helmholtz分解公式结合使用了新的符合恢复策略。产生的后验误差绑定涉及沿元素面上的切向导数跳跃。许多实际情况也证明了局部下限。还提出了数值实验，突出显示了后验误差的实际值作为误差估计器。
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.
