An interface formulation of the Laplace-Beltrami problem on piecewise smooth surfaces

TL;DR

This paper develops an interface formulation and $L^2$-invertibility theory for the Laplace-Beltrami problem on piecewise smooth surfaces, enabling accurate numerical solutions relevant to physics applications like fluid dynamics and electromagnetics.

Contribution

It extends weak formulations and integral equation approaches to piecewise smooth surfaces and reformulates the problem as an interface problem with continuity conditions.

Findings

High-order numerical examples validate the analysis.

The approach extends to general 3D embedded surfaces.

Supports applications in physics like surface diffusion and electromagnetics.

Abstract

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 develop $L^2$-invertibility theory for the Laplace-Beltrami operator on piecewise smooth surfaces, extending earlier weak formulations and integral equation approaches on smooth surfaces. Furthermore, we reformulate the weak form of the problem as an interface problem with continuity conditions across edges of adjacent piecewise smooth panels of the surface. 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.