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.
翻译:极小元素方法是众所周知的,可以承认在满足最大角条件的简化模类模类中,最优化地趋同于满足最大角条件。但文献中不知道如何将这一条件概括为聚赫德拉。在这项工作中,我们争辩说,虚拟元素方法(Vems)有可能采用这一代。特别是,我们为VEMs开发了一种厌食性分析框架,虚拟空间和投影空间仍然抽象,并且可以适应问题,从而推进VEMs的“虚拟”精神。在此框架下,将分析三个厌食性案例:(1) 元素仅含有非刻录的球,但不一定是这些球的恒星锥;(2) 元素从背景的Cartesian mesh 中任意切割,这种元素可能极为压缩;(3) 元素含有虚拟空间涉及不连续系数的不同材料;保证错误估计独立于多面元素形状。目前的工作将大大改进文献中目前的理论结果,并扩大了VEMs的应用范围。