The ultraspherical spectral element method

•A sparse spectral element method in the high-polynomial regime.•Fast direct solver allows hp-adaptivity to be efficiently performed.•Precomputed operators can be reused for fast solves in implicit time-stepping.•Open-source software system, ultraSEM, for flexible spectral element computations. We introduce a novel spectral element method based on the ultraspherical spectral method and the hierarchical Poincaré–Steklov scheme for solving second-order linear partial differential equations on polygonal domains with unstructured quadrilateral or triangular meshes. Properties of the ultraspherical spectral method lead to almost banded linear systems, allowing the element method to be competitive in the high-polynomial regime (p>5). The hierarchical Poincaré–Steklov scheme enables precomputed solution operators to be reused, allowing for fast elliptic solves in implicit and semi-implicit time-steppers. The resulting spectral element method achieves an overall computational complexity of O(p4/h3) for mesh size h and polynomial order p, enabling hp-adaptivity to be efficiently performed. We develop an open-source software system, ultraSEM, for flexible, user-friendly spectral element computations in MATLAB.