Tuesday, March 18, 2008
Mathematical Sciences Colloquium
Speaker: Jie Shen (Purdue University) "Fast Spectral-Galerkin Methods: Algorithms, Analysis and Applications"

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 a large class of 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 present applications of these fast spectral-Galerkin algorithms to a number of challenging scientific and engineering problems, such as high-order nonlinear wave equations, bounded obstacle scattering with medium to high frequencies and multiphase incompressible flows including liquid crystal flows.