High-order Galerkin method for Helmholtz and Laplace problems on multiple open arcs

Jerez-Hanckes, Carlos; Pinto, José

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 6, pp. 1975-2009.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

We present a spectral Galerkin numerical scheme for solving Helmholtz and Laplace problems with Dirichlet boundary conditions on a finite collection of open arcs in two-dimensional space. A boundary integral method is employed, giving rise to a first kind Fredholm equation whose variational form is discretized using weighted Chebyshev polynomials. Well-posedness of the discrete problems is established as well as algebraic or even exponential convergence rates depending on the regularities of both arcs and excitations. Our numerical experiments show the robustness of the method with respect to number of arcs and large wavenumber range. Moreover, we present a suitable compression algorithm that further accelerates computational times.

Classification : 65R20, 65N22, 65N35, 65N38
Mots-clés : Boundary integral equations, spectral methods, wave scattering problems, screens problems, non-Lipschitz domains

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