We extend the nonconforming Trefftz virtual element method introduced in arXiv:1805.05634 to the case of the fluid-fluid interface problem, that is, a Helmholtz problem with piecewise constant wave number. With respect to the original approach, we address two additional issues: firstly, we define the coupling of local approximation spaces with piecewise constant wave numbers, secondly, we enrich such local spaces with special functions capturing the physical behaviour of the solution to the target problem. As these two issues are directly related to an increase of the number of degrees of freedom, we use a reduction strategy inspired by arXiv:1807.11237, which allows to mitigate the growth of the dimension of the approximation space when considering $h$- and $p$-refinements. This renders the new method highly competitive in comparison to other Trefftz and quasi-Trefftz technologies tailored for the Helmholtz problem with piecewise constant wave number. A wide range of numerical experiments, including the $p$-version with quasi-uniform meshes and the $hp$-version with isotropic and anisotropic mesh refinements, is presented.

中文翻译：

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非一致 Trefftz 虚元法在分段恒定波数的 Helmholtz 问题上的推广

我们将 arXiv:1805.05634 中引入的非一致性 Trefftz 虚拟元方法扩展到流体-流体界面问题的情况，即具有分段恒定波数的亥姆霍兹问题。关于原始方法，我们解决了两个额外的问题：首先，我们定义了局部近似空间与分段常数波数的耦合，其次，我们用捕获目标问题解的物理行为的特殊函数来丰富这些局部空间. 由于这两个问题与自由度数的增加直接相关，因此我们使用了受 arXiv:1807.11237 启发的缩减策略，当考虑 $h$- 和 $ p$-改进。与为具有分段恒定波数的亥姆霍兹问题量身定制的其他 Trefftz 和准 Trefftz 技术相比，这使得新方法具有很强的竞争力。介绍了各种数值实验，包括具有准均匀网格的 $p$-版本和具有各向同性和各向异性网格细化的 $hp$-版本。