A boundary integral equation method for the transmission eigenvalue problem

Cakoni, Fioralba; Kreß, Rainer

doi:10.1080/00036811.2016.1189537

Cited by 34 publications

(48 citation statements)

References 25 publications

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“…It can be seen to be valid for

E \in L_{{curl}^{2}}^{2} false(D false)

with the boundary values interpreted in the sense of the traces

γ E \in H_{normalt}^{- \frac{1}{2}} false(\partial D false)

and

γ curl E \in H_{normalt}^{- \frac{3}{2}} false(\partial D false)

as introduced in the previous section. Since for divergence free

E \in L_{{curl}^{2}}^{2} false(D false)

, we have Δ E ∈ L 2 ( D ), the trace

ν \cdot E \in H_{normalt}^{- \frac{1}{2}} false(\partial D false)

also is well defined (see Cakoni and Kress). We will use to establish the following regularity properties.…”

Section: Vector Potential Theory Revisitedmentioning

confidence: 99%

“…For the contour integral in Beyn's method (see Cakoni and Kress and Beyn), we chose ellipses

\partial Ω = \{\frac{k_{m a x} + k_{m i n}}{2} + \frac{k_{m a x} - k_{m i n}}{2} \cos t + i β \sin t : t \in [0, 2 π]\}

and used 128 quadrature points in the composite trapezoidal rule. Here, ( k m i n , k m a x ) is the interval in which we are searching for the transmission eigenvalues and β =0.01 corresponds to the minor axis of the ellipse that is chosen rather small.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…In this paper, we extend the boundary integral equation approach that we developed in Cakoni and Kress for the transmission eigenvalue problem for the Helmholtz equation in the case of a constant refractive index. Boundary integral equation methods were first used in the context of transmission eigenvalues for the Helmholtz problem by Cossonnière and Haddar .…”

Section: Introductionmentioning

confidence: 99%

“…In their work, they used Green's representation formula to derive a system of two linear boundary integral equations that are equivalent to the transmission eigenvalue problem and depend nonlinearly on the eigenvalue parameter k . Using parts of the analysis in Cossonnière and Haddar, we were able to develop a new formulation that leads to only one linear boundary integral equation in terms of a Dirichlet‐to‐Neumann operator and used it also for numerical computations of transmission eigenvalues (see Cakoni and Kress). The analysis in Cossonnière and Haddar was extended by Cakoni et al to the transmission eigenvalue problem for the Maxwell equations leading again to a system of two linear integral equations.…”

Section: Introductionmentioning

confidence: 99%

“…The analysis in Cossonnière and Haddar was extended by Cakoni et al to the transmission eigenvalue problem for the Maxwell equations leading again to a system of two linear integral equations. It is the purpose of this paper to extend our approach from Cakoni and Kress to again obtain a formulation with only one integral equation.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A boundary integral equation for the transmission eigenvalue problem for Maxwell equation

Cakoni

Yaman

Kreß

et al. 2018

Math Methods in App Sciences

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We propose a new integral equation formulation to characterize and compute transmission eigenvalues in electromagnetic scattering. As opposed to the approach that was recently developed by Cakoni, Haddar and Meng (2015) which relies on a two‐by‐two system of boundary integral equations, our analysis is based on only one integral equation in terms of the electric‐to‐magnetic boundary trace operator that results in a simplification of the theory and in a considerable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further, we use the numerical algorithm for analytic nonlinear eigenvalue problems that was recently proposed by Beyn (2012) for the numerical computation of the transmission eigenvalues via this new integral equation.

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“…It can be seen to be valid for