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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 computational 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. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 941–955, 2017

Pseudo‐Spectral Methods for the Laplace‐Beltrami Equation and the Hodge Decomposition on Surfaces of Genus One