A Direct Solver for the Rapid Solution of Boundary Integral Equations on Axisymmetric Surfaces in Three Dimensions

TL;DR

This paper introduces a fast, accurate method for solving boundary integral equations on axisymmetric surfaces in 3D by reducing the problem to a 2D curve using Fourier transforms and high-order quadratures, enabling rapid solutions especially for multiple right-hand sides.

Contribution

The paper presents a novel direct solver that combines Fourier reduction, high-order Gaussian quadrature, and recursion for Legendre functions to efficiently solve BIEs on axisymmetric surfaces, handling non-symmetric loads.

Findings

The method achieves rapid setup and solve times for large discretizations.

It maintains high accuracy near singular kernels using modified quadrature.

The approach is effective for BIEs related to Laplace's equation on complex surfaces.

Abstract

A scheme for rapidly and accurately computing solutions to boundary integral equations (BIEs) on rotationally symmetric surfaces in three dimensions 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 used 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 Laplace's equation, 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 320 000 points, the set-up phase of the algorithm takes 2 minutes on a standard desktop, and then solves can be executed in 0.5 seconds.