The talk is motivated by application to fractal antenna engineering, where antennas with self-similar shapes operate across multiple frequencies. Recent works by Chandler-Wilde, Gibbs, Hewett, Moiola and their co-workers propose boundary integral formulations for solving Helmholtz scattering problems on fractal screens with Dirichlet boundary conditions. These boundary integral equations utilize the Hausdorff measure on fractals instead of the standard Lebesgue's measure. This makes the evaluation of the boundary integral difficult from a numerical viewpoint. To tackle this point, we propose new interpolatory high-order tensor cubature formula on fractals, based on Chebyshev points on an interval. These formulas allow computing integrals of restrictions of regular functions to fractals with a high accuracy. We will discuss the construction of such cubature (in particular computation of the weights) and their properties (asymptotic behavior of weights).
Jérémy Heleine, David Lafontaine
