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The Laplace-Beltrami problem on closed surfaces embedded in three dimensionsarises in many areas of physics, including molecular dynamics (surfacediffusion), electromagnetics (harmonic vector fields), and fluid dynamics(vesicle deformation). In particular, the Hodge decomposition of vector fieldstangent to a surface can be computed by solving a sequence of Laplace-Beltramiproblems. Such decompositions are very important in magnetostatic calculationsand in various plasma and fluid flow problems. In this work we develop$L^2$-invertibility theory for the Laplace-Beltrami operator on piecewisesmooth surfaces, extending earlier weak formulations and integral equationapproaches on smooth surfaces. Furthermore, we reformulate the weak form of theproblem as an interface problem with continuity conditions across edges ofadjacent piecewise smooth panels of the surface. We then provide high-ordernumerical examples along surfaces of revolution to support our analysis, anddiscuss numerical extensions to general surfaces embedded in three dimensions.

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