A high-order accurate accelerated direct solver for acoustic scattering from surfaces

We describe an accelerated direct solver for the integral equations which model low-frequency acoustic scattering from curved surfaces. Surfaces are specified via a collection of smooth parameterizations given on triangles, a setting which generalizes the typical one of triangulated surfaces, and the integral equations are discretized via a high-order Nyström method. This allows for rapid convergence in cases in which high-order surface information is available. The high-order discretization technique is coupled with a direct solver based on the recursive construction of scattering matrices. The result is a solver which often attains O(N1.5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(N^{1.5})$$\end{document} complexity in the number of discretization nodes N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document} and which is resistant to many of the pathologies which stymie iterative solvers in the numerical simulation of scattering. The performance of the algorithm is illustrated with numerical experiments which involve the simulation of scattering from a variety of domains, including one consisting of a collection of 1,000 ellipsoids with randomly chosen semiaxes arranged in a grid, and a domain whose boundary has 12 curved edges and 8 corner points.