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High-order Galerkin Method for Helmholtz and Laplace Problems on Multiple Open Arcs

by C. Jerez-Hanckes and J. Pinto

(Report number 2018-49)

Abstract
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. Moreover, our numerical experiments show the robustness of the method with respect to number of arcs and large wavenumber range.

Keywords: Boundary integral equations, spectral methods, wave scattering problems, screens problems, non-Lipschitz domains

@Techreport{JP18_803,
  author = {C. Jerez-Hanckes and J. Pinto},
  title = {High-order Galerkin Method for Helmholtz and Laplace Problems on Multiple Open Arcs},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2018-49},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2018/2018-49.pdf },
  year = {2018}
}

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