Finite Element Methods for Maxwell's Transmission Eigenvalues

Peter Monk, Jiguang Sun
2012 SIAM Journal on Scientific Computing  
The transmission eigenvalue problem plays a critical role in the theory of qualitative methods for inhomogeneous media in inverse scattering theory. Efficient computational tools for transmission eigenvalues are needed to motivate improvements to theory, and, more importantly as part of inverse algorithms for estimating material properties. In this paper, we propose two finite element methods to compute a few lowest Maxwell's transmission eigenvalues which are of interest in applications. Since
more » ... the discrete matrix eigenvalue problem is large, sparse, and, in particular, non-Hermitian due to the fact that the problem is neither elliptic nor self-adjoint, we devise an adaptive method which combines the Arnoldi iteration and estimation of transmission eigenvalues. Exact transmission eigenvalues for balls are derived and used as a benchmark. Numerical examples are provided to show the viability of the proposed methods and to test the accuracy of recently derived inequalities for transmission eigenvalues. Introduction. The interior transmission problem (ITP) has attracted a lot of attention recently in the inverse scattering community due to its importance in the study of direct and inverse scattering problem for non-absorbing inhomogeneous media [10, 13, 22, 6, 18, 7, 11] . The ITP is a boundary value problem in a bounded domain which is neither elliptic nor self-adjoint. It is known that the associated spectrum, the so called transmission eigenvalues, closely relates to the denseness of the far field operator and plays a critical role in qualitative methods in inverse scattering, e.g., the linear sampling method and factorization method [3, 17] . Furthermore, transmission eigenvalues can be obtained from either far field or near field data and used to estimate the index of refraction of an inhomogeneous medium [4, 25] . The transmission eigenvalue problem is not directly covered by standard partial differential equation theory. Until recently, little was known about the existence of transmission eigenvalues except in the case of a spherically stratified medium [10] . In [22] , Päivärinta and Sylvester proved the existence of (real) transmission eigenvalues in the general case. Cakoni et al. [5] showed that transmission eigenvalues form a countable set with infinity as the only accumulation points. For more results on the existence of transmission eigenvalues, we refer the readers to [23, 5, 18, 7, 6] and the references therein. Note that the existence of complex transmission eigenvalues is suggested by numerical examples [11] and can be proved for spherically stratified media under certain conditions [2] . A region in the complex plane where complex eigenvalues must lie was derived in [2] for scalar problems and later in [27] for vector problems. However, for general cases, the existence of eigenvalues with non-zero imaginary part is an open problem. The need for efficient computational tools is of great importance because theoretical results are still partial and numerical evidence might lead theorists in the right direction. Moreover, numerical methods for computing transmission eigenvalues are needed in some algorithms to estimate the index of refraction [4, 25] . There are only a few papers dealing with the numerical computation of transmission eigenvalues. In *
doi:10.1137/110839990 fatcat:d3roo5575ncavlaf74xuelmzze