Abstract
AbstractSparse 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 nonclassical orthogonal polynomials on disk slices 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 (nonclassical) univariate orthogonal polynomials. We apply the framework to solving the Poisson, variable coefficient Helmholtz, and biharmonic equations. In this paper, we focus on constant Dirichlet boundary conditions, as well as zero Dirichlet and Neumann boundary conditions, with other types of boundary conditions requiring future work.
Highlights
This paper develops sparse spectral methods for solving linear partial differential equations (PDEs) on a special class of geometries that includes disk slices and trapeziums
We focus on the disk-slice, where ρ(x) = 1 − x2, (α, β) ⊂ (0, 1), and (γ, δ) = (−1, 1), and discuss an extension to other geometries in the Appendices
We show that PDEs become sparse linear systems when viewed as acting on expansions involving a family of orthogonal polynomials (OPs) that generalize Jacobi polynomials, mirroring the ultraspherical spectral method for ordinary differential equations[1] and its analogue on the disk[2] and triangle.[3,4]
Summary
This paper develops sparse spectral methods for solving linear partial differential equations (PDEs) on a special class of geometries that includes disk slices and trapeziums. We show that PDEs become sparse linear systems when viewed as acting on expansions involving a family of orthogonal polynomials (OPs) that generalize Jacobi polynomials, mirroring the ultraspherical spectral method for ordinary differential equations[1] and its analogue on the disk[2] and triangle.[3,4] On the disk-slice the family of weights we consider are of the form. By exploiting the connection with 1D OPs, we can construct discretizations of general partial differential operators of size p(p − 1)∕2 × p(p − 1)∕2 in O(p3) operations, where p is the total polynomial degree This compares favorably to O(p6) operations if one proceeds naïvely.
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