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

Abstract

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.

On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method

“…In practice, ε can not be taken as large as we wish. In [23] where the Maxwell's equations for Tellegen media is studied, ε(x) is the sum of the electric permittivity and the square of Tellegen parameter. By choosing large Tellegen parameter, we can enlarge the parameter ε(x) as we want.…”

“…In recent years, several papers have been devoted to developing efficient algorithms for computing transmission eigenvalues of 2d/3d TEP [3,11,14,16,17,20,21,22,23,24,26,27]. Three finite element methods (FEMs) and a coupled boundary element method were developed to solve the 2d/3d TEP in [11,14,17] (see the book [30] for more details).…”

“…Furthermore, a multilevel correction method was used to reduce the solution of TEP into some linear boundary value problems which could be solved by the multigrid method [18]. For results related to our current work, in [23,24], the TEP for general inhomogeneous media are discretized into QEP with symmetric coefficient matrices. For such QEP, a secant-type iteration for computing several smallest positive transmission eigenvalues accurately was proposed in [24], and a quadratic Jacobi-Davidson method with nonequivalent deflation technique for computing a large number positive transmission eigenvalues was developed in [23].…”

“…For results related to our current work, in [23,24], the TEP for general inhomogeneous media are discretized into QEP with symmetric coefficient matrices. For such QEP, a secant-type iteration for computing several smallest positive transmission eigenvalues accurately was proposed in [24], and a quadratic Jacobi-Davidson method with nonequivalent deflation technique for computing a large number positive transmission eigenvalues was developed in [23]. Remind that the index of refraction ε(x) increases to infinity as we increase the Tellegen parameter.…”

“…Numerical simulations demonstrate that the positive transmission eigenvalues are densely distributed in an interval near the origin. Similar to the method in [23], the TEP (1) is discretized into a QEP and a quadratic Jacobi-Davidson method with nonequivalence deflation is applied to compute a large number of positive eigenvalues. It turns out in the case here there exists an eigenvalue-free interval near the origin.…”