  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.
