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

Sheehan Olver

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.

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

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 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.
Paper Structure (11 sections, 1 theorem, 179 equations, 2 tables)

This paper contains 11 sections, 1 theorem, 179 equations, 2 tables.

Table of Contents

  1. Invariance and Equivariance
  2. Symmetry-adapted bases
  3. Vector calculus
  4. Vector Zernike polynomials
  5. Orthogonal polynomials for a weight that is normal at the boundary
  6. Expressions for differential operators.
  7. Exactness of the resulting de Rham sub-complexes.
  8. Cylinders with periodic boundary conditions
  9. Exactness of the periodic cylinder de Rham complex with boundary conditions.
  10. Finite cylinders
  11. Exactness of the finite cylinder de Rham complex with boundary conditions.

Key Result

theorem 1

Suppose we have an LU factorisation: where $L$ and $U$ are invertible. Vector OPs with respect to $M V$ in ${\mathcal{M}}$ are given by the columns of ${\bf P}^V U^{-1}$.

Theorems & Definitions (19)

  • Remark 1
  • proof
  • proof
  • proof
  • proof
  • Remark 2
  • proof
  • Remark 3
  • proof
  • Remark 4
  • ...and 9 more