Boundary element approximation for Maxwell's eigenvalue problem

Abstract

We introduce a new method for computing eigenvalues of the Maxwell operator with boundary finite elements. On bounded domains with piecewise constant material coefficients, the Maxwell solution for fixed wave number can be represented by boundary integrals, which allows to reduce the eigenvalue problem to a nonlinear problem for determining the wave number along with boundary and interface traces. A Galerkin discretization yields a smooth nonlinear matrix eigenvalue problem that is solved by Newton's method or, alternatively, the contour integral method. Several numerical results including an application to the band structure computation of a photonic crystal illustrate the efficiency of this approach. Copyright © 2013 John Wiley & Sons, Ltd.

Solving Maxwell's eigenvalue problem via isogeometric boundary elements and a contour integral method

“…However, for the specific densities of resonant modes within interior cavities, this smoothness assumption will often be fulfilled. For the cube, an analytical representation of is known to be smooth, 16 and even for other, non‐trivial geometries this can be argued.…”

“…A first approach to a boundary element eigenvalue problem via the contour integral method was investigated in Wieners and Xin, 16 however neither with a higher order approach, a discussion of the related convergence theory, or within the isogeometric setting.…”

The nonlinear eigenvalue problem

“…This method has been applied successfully, e.g. for the solution of NEPs arising from resonance problems related to fluid–solid interaction (Kimeswenger, Steinbach and Unger 2014) and Maxwell eigenvalue problems (Wieners and Xin 2013). The latter reference also contains numerical comparisons of the integral approach with a Newton method. For and we obtain the ‘Integral algorithm 2’ of Beyn (2012).…”

Higher-order wavelet reconstruction/differentiation filters and Gibbs phenomena

“…For highly symmetric cases such as spheroidal and ellipsoidal nanoparticles, analytical Mie theory and vector harmonics may be used [31,32]. For more realistic nanoparticles, numerical approaches can be taken such as the FDTD [13,14], finite-element [33], boundary-element [34] and discrete dipole approximation [35] methods. Wavelet-based approaches have also been developed, including the multiresolution time domain (MRTD) [36,37] and wavelet collocation [38,39] methods.…”