The ultraspherical spectral element method

TL;DR

This paper presents a new ultraspherical spectral element method combining hierarchical Poincaré-Steklov schemes for efficient high-order PDE solutions on complex polygonal domains, with an open-source MATLAB implementation.

Contribution

It introduces a novel spectral element method that leverages ultraspherical spectral techniques and hierarchical Poincaré-Steklov schemes for efficient PDE solving on unstructured meshes.

Findings

Achieves almost banded linear systems for high polynomial degrees

Enables fast elliptic solves with precomputed solution operators

Provides an open-source MATLAB software for spectral element computations

Abstract

We introduce a novel spectral element method based on the ultraspherical spectral method and the hierarchical Poincar\'{e}-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 Poincar\'{e}-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 $\mathcal{O}(p^4/h^3)$ 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.