Second-kind integral equations for the Laplace-Beltrami problem on surfaces in three dimensions
This work provides a well-conditioned numerical method for solving the Laplace-Beltrami equation on surfaces, benefiting applications in physics, geometry, and machine learning.
The authors present novel second-kind integral equations for the Laplace-Beltrami problem on surfaces, eliminating the need for an approximate in-surface Green's function. Numerical examples on curvilinear triangle surfaces demonstrate accuracy and stability.
The Laplace-Beltrami problem $Δ_Γψ= f$ 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.