Fast integral equation methods for the Laplace-Beltrami equation on the sphere

TL;DR

This paper introduces an efficient O(N) integral equation method for solving the Laplace-Beltrami equation on the sphere, utilizing stereographic projection and the fast multipole method for rapid computations.

Contribution

The paper develops a novel fast integral equation approach for the Laplace-Beltrami equation on the sphere, combining stereographic projection with the FMM for efficient solutions.

Findings

Achieves O(N) computational complexity.

Demonstrates high accuracy on multiple examples.

Effective for complex geometries with multiple islands.

Abstract

Integral equation methods for solving the Laplace-Beltrami equation on the unit sphere in the presence of multiple "islands" are presented. The surface of the sphere is first mapped to a multiply-connected region in the complex plane via a stereographic projection. After discretizing the integral equation, the resulting dense linear system is solved iteratively using the fast multipole method for the 2D Coulomb potential in order to calculate the matrix-vector products. This numerical scheme requires only O(N) operations, where $N$ is the number of nodes in the discretization of the boundary. The performance of the method is demonstrated on several examples.