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Sparse spectral methods for solving partial differential equations have beenderived in recent years using hierarchies of classical orthogonal polynomialson intervals, disks, and triangles. In this work we extend this methodology toa hierarchy of non-classical orthogonal polynomials on disk slices (e.g. ahalf-disk) and trapeziums. This builds on the observation that sparsity isguaranteed due to the boundary being defined by an algebraic curve, and thatthe entries of partial differential operators can be determined using formulaein terms of (non-classical) univariate orthogonal polynomials. We apply theframework to solving the Poisson, variable coefficient Helmholtz, andBiharmonic equations.

Sparse spectral and p-finite element methods for partial differential equations on disk slices and trapeziums