Sparse spectral methods for partial differential equations on spherical caps. Issue 1 (20th November 2021)

• Record Type: Journal Article
• Authors: Snowball, Ben; Olver, Sheehan
• Is Part Of: Transactions of mathematics and its applications. Volume 5:Issue 1(2021)
• Publication Date: 2021-11-20
• Subjects: Applied mathematics -- Periodicals; 519.05
• DOI: 10.1093/imatrm/tnab001
• Languages: English
• ISSNs: 2398-4945

Abstract

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