The ultraspherical spectral method

Alex Townsend
Cornell University

Pseudospectral methods, based on high degree polynomials, have spectral accuracy when solving differential equations but typically lead to dense and ill-conditioned matrices. The ultraspherical spectral method is a numerical technique to solve ordinary and partial differential equations, leading to almost banded well-conditioned linear systems while maintaining spectral accuracy. In this talk, we introduce the ultraspherical spectral method and develop it into a spectral element method using a modification to a hierarchical Poincaré-Steklov domain decomposition method.