Abstract
Two recently derived integral equations for the Maxwell transmission problem are compared through numerical tests on simply connected axially symmetric domains for non-magnetic materials. The winning integral equation turns out to be entirely free from false eigenwavenumbers for any passive materials, also for purely negative permittivity ratios and in the static limit, as well as free from false essential spectrum on non-smooth surfaces. It also appears to be numerically competitive to all other available integral equation reformulations of the Maxwell transmission problem, despite using eight scalar surface densities.
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Open access funding provided by Lund University. This work was supported by the Swedish Research Council under contract 2015-03780.
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Communicated by: Gunnar J Martinsson
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Appendices
Appendix: A. The entries of E k
The entries of the matrix Ek of (??) involve two families of integral operators denoted Sk and Kk. A given member in an operator family, \(S_{k}^{\alpha }\) or \(K_{k}^{\alpha }\), is defined by its superscript α, which can be a constant, a unit vector, a scalar product of unit vectors, or a cross product of unit vectors. Specifically we have for \({S_{k}^{1}}\) acting on a general density g
for \(S_{k}^{\boldsymbol u\cdot \boldsymbol v^{\prime }}\), where u and v are unit vectors,
and for \(K_{k}^{\boldsymbol u}\) and \(K_{k}^{\boldsymbol u\times \boldsymbol v^{\prime }}\)
When coding, particularly when Γ is axially symmetric and when azimuthal Fourier transforms are to be implemented, it is helpful to have \(S_{k}^{\alpha }\) and \(K_{k}^{\alpha }\) in the form
where \(s_{\alpha }(\boldsymbol r,\boldsymbol r^{\prime })\) and \(d_{\alpha }(\boldsymbol r,\boldsymbol r^{\prime })\) are static kernel factors expressed in terms of quantities introduced in (??)-(??) and the azimuthal angle 𝜃.
Here follow \(s_{\alpha }(\boldsymbol r,\boldsymbol r^{\prime })\) for the ten operators \(S_{k}^{\alpha }\) of Ek
Here follow \(d_{\alpha }(\boldsymbol r,\boldsymbol r^{\prime })\) for the 15 operators \(K_{k}^{\alpha }\) of Ek
Appendix B. Layer potentials for field evaluations
The expressions for field evaluation (??–??) involve layer potentials \(\tilde S_{k}^{\alpha }\) and \(\tilde K_{k}^{\alpha }\) which are defined analogously to the operators \(S_{k}^{\alpha }\) and \(K_{k}^{\alpha }\) in Appendix A. The only difference is that \(\tilde S_{k}^{\alpha } g(\boldsymbol r)\) and \(\tilde K_{k}^{\alpha } g(\boldsymbol r)\) have r ∈Ω−∪Ω+ while \(S_{k}^{\alpha } g(\boldsymbol r)\) and \(K_{k}^{\alpha } g(\boldsymbol r)\) have r ∈Γ. Therefore \(\tilde K_{k}^{\alpha }\) does not need the principal value.
When coding, it is helpful to have \(\tilde S_{k}^{\alpha }\) and \(\tilde K_{k}^{\alpha }\) in the form
where \(s_{\alpha }(\boldsymbol r,\boldsymbol r^{\prime })\) and \(d_{\alpha }(\boldsymbol r,\boldsymbol r^{\prime })\) are static kernel factors, some of which are already listed Appendix A. The static kernel factors needed, not listed in Appendix A, are
and
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Helsing, J., Karlsson, A. & Rosén, A. Comparison of integral equations for the Maxwell transmission problem with general permittivities. Adv Comput Math 47, 76 (2021). https://doi.org/10.1007/s10444-021-09904-4
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DOI: https://doi.org/10.1007/s10444-021-09904-4
Keywords
- Maxwell’s equations
- Electromagnetic scattering
- Transmission problem
- Boundary integral equation
- Surface plasmon wave
- Non-smooth object