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An interface formulation 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 develop
-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