Sparse spectral methods for solving differential equations

Abstract: Spectral and pseudospectral methods for solving differential equations provide memory-minimizing accuracy, and typically rapid convergence with increasing resolution. However, spectral collocation methods historically favored for their flexibility suffer from dense and poorly conditioned matrices. In this talk I will provide a general introduction to solving ordinary and partial differential equations with spectral methods. I will then introduce recently developed sparse, well-conditioned ultraspherical spectral methods that address many of the drawbacks of previous approaches.