3,118 research outputs found
Recursive integral method for transmission eigenvalues
Recently, a new eigenvalue problem, called the transmission eigenvalue
problem, has attracted many researchers. The problem arose in inverse
scattering theory for inhomogeneous media and has important applications in a
variety of inverse problems for target identification and nondestructive
testing. The problem is numerically challenging because it is non-selfadjoint
and nonlinear. In this paper, we propose a recursive integral method for
computing transmission eigenvalues from a finite element discretization of the
continuous problem. The method, which overcomes some difficulties of existing
methods, is based on eigenprojectors of compact operators. It is
self-correcting, can separate nearby eigenvalues, and does not require an
initial approximation based on some a priori spectral information. These
features make the method well suited for the transmission eigenvalue problem
whose spectrum is complicated. Numerical examples show that the method is
effective and robust.Comment: 18 pages, 8 figure
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
A spectral projection method for transmission eigenvalues
In this paper, we consider a nonlinear integral eigenvalue problem, which is
a reformulation of the transmission eigenvalue problem arising in the inverse
scattering theory. The boundary element method is employed for discretization,
which leads to a generalized matrix eigenvalue problem. We propose a novel
method based on the spectral projection. The method probes a given region on
the complex plane using contour integrals and decides if the region contains
eigenvalue(s) or not. It is particularly suitable to test if zero is an
eigenvalue of the generalized eigenvalue problem, which in turn implies that
the associated wavenumber is a transmission eigenvalue. Effectiveness and
efficiency of the new method are demonstrated by numerical examples.Comment: The paper has been accepted for publication in SCIENCE CHINA
Mathematic
Essential spectrum of local multi-trace boundary integral operators
Considering pure transmission scattering problems in piecewise constant
media, we derive an exact analytic formula for the spectrum of the
corresponding local multi-trace boundary integral operators in the case where
the geometrical configuration does not involve any junction point and all wave
numbers equal. We deduce from this the essential spectrum in the case where
wave numbers vary. Numerical evidences of these theoretical results are also
presented
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