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arXiv · 2606.13482

A Stabilized Multilevel B-Spline-Based Fast Integral Method for the Solution of the Electric Field Integral Equation

Danijel Jukić·Bernd Hofmann·Thomas F. Eibert·Simon B. Adrian
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Abstract

We present a multilevel B-spline-based fast integral method for the solution of the electric field integral equation (EFIE), combining fast Fourier transformation (FFT)-compatible kernel interpolation with robust high-order interpolation. Existing FFT-accelerated global Lagrange-based approaches rely on equidistant interpolation points and can, therefore, suffer from Runge-type instabilities at high interpolation orders, limiting robust high-accuracy compression. In contrast, B-splines on equidistant knot vectors overcome these instabilities and enable robust high-order interpolation for accurate matrix compression. Replacing Lagrange interpolation by B-spline interpolation is, however, non-trivial: B-spline coefficients do not coincide with function values at the interpolation points, and the associated sampling matrices can become ill-conditioned. To address these challenges, we introduce a knot-removal stabilization strategy, combined with exact interlevel transfers based on knot insertion, yielding accurate, well-conditioned multilevel interpolation. Moreover, we propose a factorization strategy that preserves the null space of the scalar potential operator up to machine precision and is compatible with low-frequency preconditioning techniques. Numerical results for both canonical and realistic geometries demonstrate robust high-order interpolation without the breakdown observed for Lagrange-based approaches and confirm $\mathcal{O}(N)$ complexity.

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Danijel Jukić, Bernd Hofmann, Thomas F. Eibert, Simon B. Adrian. 2026-06-11. A Stabilized Multilevel B-Spline-Based Fast Integral Method for the Solution of the Electric Field Integral Equation. https://arxiv.org/abs/2606.13482

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