Extension of the nonconforming Trefftz virtual element method to the Helmholtz problem with piecewise constant wave number
This work provides a more efficient numerical method for solving fluid-fluid interface problems in wave propagation, which is important for applications in acoustics and electromagnetics.
The paper extends the nonconforming Trefftz virtual element method to solve the Helmholtz problem with piecewise constant wave number, introducing coupling of local spaces and enrichment with special functions, while using a reduction strategy to control degrees of freedom. Numerical experiments demonstrate high competitiveness compared to other Trefftz methods.
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.