Sparsity comparison of polytopal finite element methods

TL;DR

This paper compares the efficiency and sparsity of Virtual Element and Discontinuous Galerkin methods, including hybrid and Trefftz variants, for solving PDEs on polytopal meshes, highlighting their computational trade-offs.

Contribution

It provides a comparative analysis of the sparsity and computational costs of VEM, hybrid DG, and Trefftz DG methods across various polytopal grid configurations.

Findings

VEM is conforming and generalizes finite elements to polytopal meshes.

Hybrid DG reduces costs by facet variable introduction.

Trefftz DG achieves complexity reduction with specialized basis functions.

Abstract

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.