David Mora, Iván Velásquez:
Virtual elements for the transmission eigenvalue problem on polytopal meshes
The transmission eigenvalue problem is a challenging model in the inverse scattering theory and has important applications in this topic. The aim of this paper is to analyze a $C^1$ Virtual Element Method (VEM) on polytopal meshes in $\R^d$ $(d=2,3)$ for solving a quadratic and non-self-adjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. Optimal order error estimates for the eigenfunctions and a double order for the eigenvalues are obtained by using the approximation theory for compact non-self-adjoint operators. Finally, a set of numerical tests illustrating the good performance of the virtual scheme are presented.