# Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds

## TL;DR

This paper introduces spectral exterior calculus methods for solving PDEs on radial manifolds, achieving high-precision discretizations of differential operators using spherical harmonics and Lebedev quadrature.

## Contribution

It develops a spectral exterior calculus framework with hyperinterpolation techniques for accurate numerical solutions on radial manifolds, including convergence analysis and applications to Laplace-Beltrami equations.

## Key findings

- Spectral convergence of numerical exterior derivative and Hodge star operators.
- High-precision discretizations achieved with spherical harmonics and Lebedev quadrature.
- Successful application to Laplace-Beltrami equations on radial manifolds.

## Abstract

We develop exterior calculus approaches for partial differential equations on radial manifolds. We introduce numerical methods that approximate with spectral accuracy the exterior derivative $\mathbf{d}$, Hodge star $\star$, and their compositions. To achieve discretizations with high precision and symmetry, we develop hyperinterpolation methods based on spherical harmonics and Lebedev quadrature. We perform convergence studies of our numerical exterior derivative operator $\overline{\mathbf{d}}$ and Hodge star operator $\overline{\star}$ showing each converge spectrally to $\mathbf{d}$ and $\star$. We show how the numerical operators can be naturally composed to formulate general numerical approximations for solving differential equations on manifolds. We present results for the Laplace-Beltrami equations demonstrating our approach.