Virtual element method for the Helmholtz transmission eigenvalue problem of anisotropic media

In this paper, we propose a conforming virtual element method for the Helmholtz transmission eigenvalue problem of anisotropic media. By using T-coercivity theory, the spectral approximation theory of compact operator and the projection and interpolation error estimates, we prove the spectral convergence of the discrete scheme and the optimal a priori error estimates for the discrete eigenvalues and eigenfunctions. The virtual element method has great flexibility in handling polygonal meshes, which motivates us to construct a fully computable a posteriori error estimator for the virtual element method. Then the upper bound of the approximation error is derived from the residual equation and the inf-sup condition. In turn, the related lower bound is established by using the bubble function strategy. Finally, we provide numerical examples to verify the theoretical results, including the optimal convergence of the virtual element scheme on uniformly refined meshes and the efficiency of the estimator on adaptively refined meshes.