Sparsity comparison of polytopal finite element methods

Abstract

In this work we compare crucial parameters for efficiency of different finite element methods for solving partial differential equations 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 transferred 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.