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