# A numerical study of the dispersion and dissipation properties of virtual element methods for the Helmholtz problem

## TL;DR

This paper numerically investigates the dispersion and dissipation characteristics of virtual element methods for the Helmholtz problem, comparing conforming and nonconforming approaches and benchmarking against other numerical methods.

## Contribution

It provides a comparative numerical analysis of virtual element methods, highlighting how conformity and Trefftz basis functions influence dispersion and dissipation properties.

## Key findings

- Dispersion and dissipation depend on conformity and Trefftz basis use.
- Nonconforming Trefftz virtual element method shows distinct behavior from conforming methods.
- Results compared with plane wave discontinuous Galerkin and polynomial finite element methods.

## Abstract

We study numerically the dispersion and dissipation properties of the plane wave virtual element method and the nonconforming Trefftz virtual element method for the Helmholtz problem. Whereas the former method is based on a conforming virtual partition of unity approach in the sense that the local (implicitly defined) basis functions are given as modulations of lowest order harmonic virtual element functions with plane waves, the latter one represents a pure Trefftz method with local edge-related basis functions that are eventually glued together in a nonconforming fashion. We will see that the qualitative and quantitative behavior of dissipation and dispersion of the method hinges upon the level of conformity and the use of Trefftz basis functions. To this purpose, we also compare the results to those obtained in [15] for the plane wave discontinuous Galerkin method, and to those for the standard polynomial based finite element method.

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Source: https://tomesphere.com/paper/1906.09965