Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds
For researchers solving PDEs on manifolds, this provides a high-precision numerical framework, though it is an incremental extension of existing spectral methods to radial manifolds.
The paper develops spectral numerical methods for exterior calculus on radial manifolds, achieving spectral convergence for the exterior derivative and Hodge star operators, and demonstrates their application to Laplace-Beltrami equations.
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.