Pseudo-spectral methods for the Laplace-Beltrami equation and the Hodge decomposition on surfaces of genus one

The inversion of the Laplace-Beltrami operator and the computation of the Hodge decomposition of a tangential vector field on smooth surfaces arise as computational tasks in many areas of science, from computer graphics to machine learning to com- putational physics. Here, we present a high-order accurate pseudo-spectral approach, applicable to closed surfaces of genus one in three dimensional space, with a view toward applications in plasma physics and fluid dynamics.