Sparse spectral methods for partial differential equations on spherical caps

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.