Second-kind integral equations for the Laplace-Beltrami problem on surfaces in three dimensions

TL;DR

This paper introduces new second-kind integral equations for solving the Laplace-Beltrami problem on 3D surfaces, avoiding complex parametrizations and enabling efficient numerical solutions with standard fast algorithms.

Contribution

It develops a novel integral equation formulation that does not require constructing a parametrix, improving stability and computational efficiency for surface Laplace-Beltrami problems.

Findings

The integral equations are well-conditioned and compatible with fast multipole methods.

Numerical examples demonstrate high accuracy and stability.

The approach simplifies solving Laplace-Beltrami problems on complex surfaces.

Abstract

The Laplace-Beltrami problem $\Delta_\Gamma \psi = f$ has several applications in mathematical physics, differential geometry, machine learning, and topology. In this work, we present novel second-kind integral equations for its solution which obviate the need for constructing a suitable parametrix to approximate the in-surface Green's function. The resulting integral equations are well-conditioned and compatible with standard fast multipole methods and iterative linear algebraic solvers, as well as more modern fast direct solvers. Using layer-potential identities known as Calder\'on projectors, the Laplace-Beltrami operator can be pre-conditioned from the left and/or right to obtain second-kind integral equations. We demonstrate the accuracy and stability of the scheme in several numerical examples along surfaces described by curvilinear triangles.