195 research outputs found
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
Revisiting the Jones eigenproblem in fluid-structure interaction
The Jones eigenvalue problem first described by D.S. Jones in 1983 concerns
unusual modes in bounded elastic bodies: time-harmonic displacements whose
tractions and normal components are both identically zero on the boundary. This
problem is usually associated with a lack of unique solvability for certain
models of fluid-structure interaction. The boundary conditions in this problem
appear, at first glance, to rule out {\it any} non-trivial modes unless the
domain possesses significant geometric symmetries. Indeed, Jones modes were
shown to not be possible in most domains (see article by T. Harg\'e
1990). However, we should in this paper that while the existence of Jones modes
sensitively depends on the domain geometry, such modes {\it do} exist in a
broad class of domains. This paper presents the first detailed theoretical and
computational investigation of this eigenvalue problem in Lipschitz domains. We
also analytically demonstrate Jones modes on some simple geometries
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
Extended Sampling Method in Inverse Scattering
A new sampling method for inverse scattering problems is proposed to process
far field data of one incident wave. As the linear sampling method, the method
sets up ill-posed integral equations and uses the (approximate) solutions to
reconstruct the target. In contrast, the kernels of the associated integral
operators are the far field patterns of sound soft balls. The measured data is
moved to right hand sides of the equations, which gives the method the ability
to process limit aperture data. Furthermore, a multilevel technique is employed
to improve the reconstruction. Numerical examples show that the method can
effectively determine the location and approximate the support with little a
priori information of the unknown target
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