An efficient numerical algorithm for computing densely distributed positive interior transmission eigenvalues

We propose an efficient eigensolver for computing densely distributed spectrum of the two-dimensional transmission eigenvalue problem (TEP) which is derived from Maxwell’s equations with Tellegen media and the transverse magnetic mode. The discretized governing equations by the standard piecewise linear finite element method give rise to a large-scale quadratic eigenvalue problem (QEP). Our numerical simulation shows that half of the positive eigenvalues of the QEP are densely distributed in some interval near the origin. The quadratic Jacobi-Davidson method with a so-called non-equivalence deflation technique is proposed to compute the dense spectrum of the QEP. Extensive numerical simulations show that our proposed method makes the convergence efficiently even it needs to compute more than 5000 desired eigenpairs. Numerical results also illustrate that the computed eigenvalue curves can be approximated by the nonlinear functions which can be applied to estimate the denseness of the eigenvalues for the TEP.