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

TL;DR

This paper introduces a high-order pseudo-spectral method for solving the Laplace-Beltrami equation and computing the Hodge decomposition on genus-one surfaces, enabling precise analysis in physics and graphics.

Contribution

The work develops a novel pseudo-spectral approach specifically designed for genus-one surfaces, improving accuracy and efficiency in solving related geometric PDEs.

Findings

Achieves high-order accuracy in solving Laplace-Beltrami equations.

Effectively computes Hodge decomposition on toroidal surfaces.

Applicable to applications in plasma physics and fluid dynamics.

Abstract

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.