论文标题
在使用虚拟元素方法的Polyhedra的最大角度条件下
On the maximum angle conditions for polyhedra with virtual element methods
论文作者
论文摘要
有限元方法是众所周知的，可以在满足最大角度条件的简单网络上接受强大的最佳收敛。但是，文献中未知如何将这种条件概括为多面体。在这项工作中，我们认为虚拟元素方法（VEMS）可能这一代是可能的。特别是，我们为VEMS开发了一个各向异性分析框架，该框架虚拟空间和投影空间保持抽象，并且可以解决问题，从而提出了``VEM的虚拟''精神。在此框架下将分析三个各向异性病例：（1）元素仅包含不碎片的铭文球，但不一定是这些球的恒星凸； （2）元素是从背景笛卡尔网格中任意切割的，这可能会缩小； （3）元素包含不同材料，虚拟空间涉及不连续系数。误差估计值可确保与多面体元素形状无关。目前的工作在很大程度上改善了文献中当前的理论结果，并扩大了VEM的应用范围。
Finite element methods are well-known to admit robust optimal convergence on simplicial meshes satisfying the maximum angle conditions. But how to generalize this condition to polyhedra is unknown in the literature. In this work, we argue that this generation is possible for virtual element methods (VEMs). In particular, we develop an anisotropic analysis framework for VEMs where the virtual spaces and projection spaces remain abstract and can be problem-adapted, carrying forward the ``virtual'' spirit of VEMs. Three anisotropic cases will be analyzed under this framework: (1) elements only contain non-shrinking inscribed balls but are not necessarily star convex to those balls; (2) elements are cut arbitrarily from a background Cartesian mesh, which can extremely shrink; (3) elements contain different materials on which the virtual spaces involve discontinuous coefficients. The error estimates are guaranteed to be independent of polyhedral element shapes. The present work largely improves the current theoretical results in the literature and also broadens the scope of the application of VEMs.
