Wavenumber-explicit analysis for the Helmholtz h-BEM: error estimates and iteration counts for the Dirichlet problem

• Published in: Numerische Mathematik 2019-06, Vol.142 (2), p.329-357
• Main Authors: Galkowski, Jeffrey, Müller, Eike H., Spence, Euan A.
• Format: Article
• Language: English
• Subjects: Boundary element method; Curvature; Dirichlet problem; Error analysis; Galerkin method; Helmholtz equations; Integral equations; Iterative methods; Iterative solution; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical and Computational Physics; Simulation; Theoretical; Wavelengths
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-019-01032-y

We consider solving the exterior Dirichlet problem for the Helmholtz equation with the h -version of the boundary element method (BEM) using the standard second-kind combined-field integral equations. We prove a new, sharp bound on how the number of GMRES iterations must grow with the wavenumber k to have the error in the iterative solution bounded independently of k as k → ∞ when the boundary of the obstacle is analytic and has strictly positive curvature. To our knowledge, this result is the first-ever sharp bound on how the number of GMRES iterations depends on the wavenumber for an integral equation used to solve a scattering problem. We also prove new bounds on how h must decrease with k to maintain k -independent quasi-optimality of the Galerkin solutions as k → ∞ when the obstacle is nontrapping.