  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). Using classical potential theory, the Laplace-Beltrami
operator can be pre-/post-conditioned with an integral operator whose kernel is
translation invariant, resulting in well-conditioned Fredholm integral
equations of the second-kind. These equations have the standard~$1/r$ kernel
from potential theory, and therefore the equations can be solved rapidly and
accurately using a combination of fast multipole methods (FMMs) and high-order
quadrature corrections. In this work we detail such a scheme, presenting two
alternative integral formulations of the Laplace-Beltrami problem, each of
whose solution can be obtained via FMM acceleration. We then present several
applications of the solvers, focusing on the computation of what are known as
harmonic vector fields, relevant for many applications in electromagnetics. A
battery of numerical results are presented for each application, detailing the
performance of the solver in various geometries.
