Stabilization-Free Virtual Element Method for the Transmission Eigenvalue Problem on Anisotropic Media

In this paper, we develop the stabilization-free virtual element method for the Helmholtz transmission eigenvalue problem on anisotropic media. The eigenvalue problem is a variable-coefficient, non-elliptic, non-selfadjoint and nonlinear model. Separating the cases of the index of refraction $n≠1$ and $n≡1,$ the stabilization-free virtual element schemes are proposed, respectively. Furthermore, we prove the spectral approximation property and error estimates in a unified theoretical framework. Finally, a series of numerical examples are provided to verify the theoretical results, show the benefits of the stabilization-free virtual element method applied to eigenvalue problems, and implement the extensions to high-order and high-dimensional cases.