A high-order Nystrom discretization scheme for boundary integral equations defined on rotationally symmetric surfaces

TL;DR

This paper introduces a fast, high-order discretization scheme for boundary integral equations on rotationally symmetric surfaces, leveraging Fourier transforms and specialized quadratures for efficiency and accuracy.

Contribution

The paper presents a novel high-order Nystrom discretization scheme that reduces BIEs on symmetric surfaces to simpler problems, enabling rapid solutions especially for multiple right-hand sides.

Findings

Efficient solution of BIEs on symmetric surfaces with high accuracy.

Significant reduction in computational time for large discretizations.

Effective handling of non-symmetric loads using the proposed method.

Abstract

A scheme for rapidly and accurately computing solutions to boundary integral equations (BIEs) on rotationally symmetric surfaces in R^3 is presented. The scheme uses the Fourier transform to reduce the original BIE defined on a surface to a sequence of BIEs defined on a generating curve for the surface. It can handle loads that are not necessarily rotationally symmetric. Nystrom discretization is used to discretize the BIEs on the generating curve. The quadrature is a high-order Gaussian rule that is modified near the diagonal to retain high-order accuracy for singular kernels. The reduction in dimensionality, along with the use of high-order accurate quadratures, leads to small linear systems that can be inverted directly via, e.g., Gaussian elimination. This makes the scheme particularly fast in environments involving multiple right hand sides. It is demonstrated that for BIEs associated with the Laplace and Helmholtz equations, the kernel in the reduced equations can be evaluated very rapidly by exploiting recursion relations for Legendre functions. Numerical examples illustrate the performance of the scheme; in particular, it is demonstrated that for a BIE associated with Laplace's equation on a surface discretized using 320,800 points, the set-up phase of the algorithm takes 1 minute on a standard laptop, and then solves can be executed in 0.5 seconds.