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

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 Calderón 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.

References in corpus (3)

• Analysis of Finite Element Methods for Vector Laplacians on Surfaces
• An adaptive fast multipole accelerated Poisson solver for complex geometries
• Pseudo-spectral methods for the Laplace-Beltrami equation and the Hodge decomposition on surfaces of genus one

Cited by in corpus (1)

• Taylor States in Stellarators: A Fast High-order Boundary Integral Solver