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.

翻译：我们从数字上研究飞机波虚拟元件法的分散和散射特性,以及Helmholtz问题不兼容的Trefftz虚拟元件法。前一种方法基于一致的虚拟分割统一法,即当地(隐含定义的)基本功能被定为用飞机波调制最低顺序调和虚拟元件功能的调制器,后一种是纯Trefftz方法,具有与边缘有关的本地功能,最终以不兼容的方式粘合在一起。我们将看到,该方法的消散和散的定性和定量行为取决于符合程度和使用Trefftz基函数。为此,我们还比较了在飞机波不连续加列金方法方面在[15]中取得的结果,以及标准以多元基定值要素法方面的结果。