Numerical solutions of partial differential equations enable a broad range of scientific
research. The Dedalus project is a flexible, open-source, parallelized computational …

We review recent advances in algorithms for quadrature, transforms, differential equations
and singular integral equations using orthogonal polynomials. Quadrature based on …

Inertial waves, which are predominantly driven by the Coriolis force, likely play an important
role in solar dynamics, and, additionally, they provide a window into the solar subsurface …

We describe fast algorithms for approximating the connection coefficients between a family
of orthogonal polynomials and another family with a polynomially or rationally modified …

The paper describes a sparse direct solver for the linear systems that arise from the
discretization of an elliptic PDE on a two-dimensional domain. The scheme decomposes the …

We develop a sparse hierarchical $ hp $-finite element method ($ hp $-FEM) for the
Helmholtz equation with rotationally invariant variable coefficients posed on a two …

We develop the ultraspherical rectangular collocation (URC) method, a collocation
implementation of the sparse ultraspherical method of Olver\& Townsend for two-point …

This paper investigates the numerical approximation of the eigenvalues of regular and
singular second-order non-self-adjoint Sturm–Liouville problems using an efficient …

Spectral methods for solving partial differential equations (PDEs) depict a high order of
convergence, which is exponential when the solution is analytic. However, their applications …

We investigate the ultraspherical spectral element method for solving second-order partial
differential equations in two dimensions. Moreover, a novel coordinate transformation is …