Sparse spectral and p-finite element methods for partial differential equations on disk slices and trapeziums

TL;DR

This paper extends sparse spectral methods to non-classical orthogonal polynomials on disk slices and trapeziums, enabling efficient solutions for PDEs like Poisson and Helmholtz equations on these geometries.

Contribution

It introduces a novel hierarchy of non-classical orthogonal polynomials for sparse spectral PDE solutions on disk slices and trapeziums, expanding previous classical polynomial frameworks.

Findings

Efficient spectral methods for PDEs on disk slices and trapeziums.

Sparsity guaranteed by algebraic boundary curves.

Successful application to Poisson, Helmholtz, and Biharmonic equations.

Abstract

Sparse spectral methods for solving partial differential equations have been derived in recent years using hierarchies of classical orthogonal polynomials on intervals, disks, and triangles. In this work we extend this methodology to a hierarchy of non-classical orthogonal polynomials on disk slices (e.g. a half-disk) and trapeziums. This builds on the observation that sparsity is guaranteed due to the boundary being defined by an algebraic curve, and that the entries of partial differential operators can be determined using formulae in terms of (non-classical) univariate orthogonal polynomials. We apply the framework to solving the Poisson, variable coefficient Helmholtz, and Biharmonic equations.