In recent years, spectral methods have become increasingly popular among computational scientists and engineers because of their superior accuracy and efficiency when properly implemented. In this talk, I shall present fast spectral-Galerkin algorithms for some prototypical partial differential equations. These spectral-Galerkin algorithms have computational complexities which are comparable to those of finite difference and finite element algorithms, yet they are capable of providing much more accurate results with a significantly smaller number of unknowns. A key ingredient for the efficiency and stability of the spectral-Galerkin algorithms is to use (properly defined) generalized Jacobi polynomials as basis functions
I shall illustrate applications of these fast spectral-Galerkin algorithms to a number of scientific and engineering problems, including, nonlinear wave equations, acoustic scattering and mutli-phase flows.