In this work we compare crucial parameters for efficiency of different finite element methods for solving partial differential equations (PDEs) on polytopal meshes. We consider the Virtual Element Method (VEM) and different Discontinuous Galerkin (DG) methods, namely the Hybrid DG and Trefftz DG methods. The VEM is a conforming method, that can be seen as a generalization of the classic finite element method to arbitrary polytopal meshes. DG methods are non-conforming methods that offer high flexibility, but also come with high computational costs. Hybridization reduces these costs by introducing additional facet variables, onto which the computational costs can be transfered to. Trefftz DG methods achieve a similar reduction in complexity by selecting a special and smaller set of basis functions on each element. The association of computational costs to different geometrical entities (elements or facets) leads to differences in the performance of these methods on different grid types. This paper aims to compare the dependency of these approaches across different grid configurations.

翻译：本文比较了在多面体网格上求解偏微分方程的不同有限元方法效率的关键参数。我们考虑了虚拟单元法以及不同的间断伽辽金方法，即混合间断伽辽金方法和特雷夫茨间断伽辽金方法。虚拟单元法是一种协调方法，可视为经典有限元方法向任意多面体网格的推广。间断伽辽金方法是非协调方法，具有高度灵活性，但也伴随着较高的计算成本。混合化通过引入额外的面变量来降低这些成本，计算负担可转移至这些面变量上。特雷夫茨间断伽辽金方法则通过在每个单元上选择特殊且更小的基函数集来实现类似的复杂度降低。计算成本与不同几何实体（单元或面）的关联导致这些方法在不同网格类型上的性能差异。本文旨在比较这些方法在不同网格配置下的依赖性。