On the numerical solution of the Laplace-Beltrami problem on piecewise-smooth surfaces

The Laplace-Beltrami problem on closed surfaces embedded in three dimensions arises in many areas of physics, including molecular dynamics (surface diffusion), electromagnetics (harmonic vector fields), and fluid dynamics (vesicle deformation). In particular, the Hodge decomposition of vector fields tangent to a surface can be computed by solving a sequence of Laplace-Beltrami problems. Such decompositions are very important in magnetostatic calculations and in various plasma and fluid flow problems. In this work we present an overview of the $L^2$-invertibility of the Laplace-Beltrami operator on piecewise smooth surfaces, extending earlier weak formulations and integral equation approaches on smooth surfaces. We then provide high-order numerical examples along surfaces of revolution to support our analysis, and discuss numerical extensions to general surfaces embedded in three dimensions.