Sparse spectral methods for partial differential equations on spherical caps

TL;DR

This paper extends sparse spectral methods to spherical caps using non-classical orthogonal polynomials, enabling efficient and accurate solutions for PDEs involving the spherical Laplacian and biharmonic operators.

Contribution

It introduces a novel hierarchy of multivariate orthogonal polynomials on spherical caps for spectral PDE discretization, expanding beyond classical polynomial domains.

Findings

Effective computation of discretized operators using univariate orthogonal polynomials

Spectral convergence demonstrated for spherical Laplacian and biharmonic PDEs

Extension of spectral methods to spherical cap geometries

Abstract

In recent years, sparse spectral methods for solving partial differential equations have been derived using hierarchies of classical orthogonal polynomials on intervals, disks, disk-slices and triangles. In this work we extend the methodology to a hierarchy of non-classical multivariate orthogonal polynomials on spherical caps. The entries of discretisations of partial differential operators can be effectively computed using formulae in terms of (non-classical) univariate orthogonal polynomials. We demonstrate the results on partial differential equations involving the spherical Laplacian and biharmonic operators, showing spectral convergence.