Orthogonal polynomials for the de Rham complex on the disk and cylinder

TL;DR

This paper develops new orthogonal polynomial bases on disks and cylinders that respect the de Rham complex structure and boundary conditions, facilitating analysis of vector fields with rotational symmetry.

Contribution

It introduces explicit vector orthogonal polynomials on the disk linked to Zernike polynomials, extending to cylinders with simple recurrence relations for differential operators.

Findings

Constructed vector orthogonal polynomials related to Zernike polynomials.

Established bases that decouple the de Rham complex into small sub-complexes.

Provided explicit recurrence relations for gradient, curl, and divergence.

Abstract

This paper constructs polynomial bases that capture the structure of the de Rham complex with boundary conditions in disks and cylinders (both periodic and finite) in a way that respects rotational symmetry. The starting point is explicit constructions of vector and matrix orthogonal polynomials on the unit disk that are analogous to the (scalar) generalised Zernike polynomials. We use these to build new orthogonal polynomials with respect to a matrix weight that forces vector polynomials to be normal on the boundary of the disk. The resulting weighted vector orthogonal polynomials have a simple connection to the gradient of weighted generalised Zernike polynomials, and their curl (i.e. vorticity or rot) is a constant multiple of the standard Zernike polynomials which are orthogonal with respect to $L^2$ on the disk. This construction naturally leads to bases in cylinders with simple recurrences relating their gradient, curl and divergence. These bases decouple the de Rham complex into small exact sub-complexes.