A mixed FEM for the quad-curl eigenvalue problem

The quad-curl problem arises in the study of the electromagnetic interior transmission problem and magnetohydrodynamics (MHD). In this paper, we study the quad-curl eigenvalue problem and propose a mixed method using edge elements for the computation of the eigenvalues. To the author's knowledge, it is the first numerical treatment for the quad-curl eigenvalue problem. Under suitable assumptions on the domain and mesh, we prove the optimal convergence. In addition, we show that the divergence-free condition can be bypassed. Numerical results are provided to show the viability of the method

Surface concentration of transmission eigenfunctions

The transmission eigenvalue problem is a type of non-elliptic and non-selfadjoint spectral problem that arises in the wave scattering theory when invisibility/transparency occurs. The transmission eigenfunctions are the interior resonant modes inside the scattering medium. We are concerned with the geometric rigidity of the transmission eigenfunctions and show that they concentrate on the boundary surface of the underlying domain in two senses. This substantiates the recent numerical discovery in [10] on such an intriguing spectral phenomenon of the transmission resonance. Our argument is based on generalized Weyl's law and certain novel ergodic properties of the coupled boundary layer-potential operators which are employed to analyze the generalized transmission eigenfunctions.Comment: 25 pages and comments are welcom

Weyl Type Bound on Positive Interior Transmission Eigenvalues

This paper contains a lower bound of the Weyl type on the counting function of the positive eigenvalues of the interior transmission eigenvalue problem which justifies the existence of an infinite set of positive interior transmission eigenvalues. We consider the classical transmission problem as well as the case where the inhomogeneous medium contains an obstacle. One of the essential components of the proof is an estimate for the D-t-N operator for the Helmholtz equation for positive λ that replaces the standard parameter-elliptic estimate valid outside of the positive semi-axis