Asymptotic Spectral Insights Behind Fast Direct Solvers for High-Frequency Electromagnetic Integral Equations on Non-Canonical Geometries

Abstract

Integral-equation-based fast direct solvers for electromagnetic scattering can substantially reduce computational costs, especially in the presence of multiple excitations. We recently proposed a new high-frequency fast direct solver strategy that combines preconditioning techniques with acceleration algorithms. However, the validity of this approach applied to non-canonical geometries requires further justification. In this contribution, we collect relevant semiclassical microlocal results and use them to assess the legitimacy and effectiveness of the proposed fast direct solver in the high-frequency regime.

Figures (5)

• Figure 1: Comparison between the principal symbol \ref{['eqn:Sprinc']}, the glancing symbol \ref{['eqn:Sgl_Ai']}, the approximate glancing symbol \ref{['eqn:SglancApprox']} and the exact eigenvalues $\lambda^{\mathcal{S}^k}(\xi) = \mathrm{i} \pi a /2 \,J_{a\xi}(ka)H_{a\xi}^{(1)}(ka)$ of the single-layer operator evaluated over the circular boundary at $ka = 500$: real part (top) and imaginary part (bottom).
• Figure 2: Comparison between the glancing symbol \ref{['eqn:Dgl_Ai']}, the approximate glancing symbol \ref{['eqn:DglancApprox']} and the exact eigenvalues $\lambda^{\mathcal{D}^k}(\xi) = \mathrm{i} \pi a k/4 \,\left(J_{a\xi}(ka)H_{a\xi}^{(1)}(ka)\right)^\prime$ of the double-layer operator evaluated over the circular boundary at $ka = 500$: real part (top) and imaginary part (bottom).
• Figure 3: Comparison between the principal symbol \ref{['eqn:Nprinc']}, the glancing symbol \ref{['eqn:Ngl_Ai']}, the approximate glancing symbol \ref{['eqn:NglancApprox']} and the exact eigenvalues $\lambda^{\mathcal{N}^k}(\xi) = -\mathrm{i} \pi a k^2/2 \,J^\prime_{a\xi}(ka)H_{a\xi}^{(1)\prime}(ka)$ of the hypersingular operator evaluated over the circular boundary at $ka = 500$: real part (top) and imaginary part (bottom).
• Figure 4: Comparison between the exact current (in black) and its approximation around $s_0$\ref{['eqn:Jzapprox']} (in red) for different plane wave incidence directions ($kL/(2\pi)=80$).
• Figure 5: Comparison between the exact current (in black) and its approximation around $s_0$\ref{['eqn:Jtapprox']} (in red) for different plane wave incidence directions ($kL/(2\pi)=80$).