In recent years, spectral methods have become a major tool for computational scientists and engineers because of their superior accuracy and efficiency when properly implemented. In this talk, I shall present essential ingredients to construct fast spectral algorithms and to carry out their error analysis. I shall also present, as examples of applications, fast spectral methods for acoustic/electromagnetic scattering, microstructure evolution, mutli-phase complex  flows, as well as fractional differential equations.

Prof. Shen received his B.S. in Computational Mathematics from Peking University in 1982, and his Ph.D in Numerical Analysis from Universite de Paris-Sud at Orsay in 1987. Before joining the Purdue Faculty in Fall 2002, he served as Professor of Mathematics at Penn State University and University of Central Florida. Since Jan. 2012 he serves as the Director of Center for Computational and Applied Mathematics at Purdue University. He is a recipient of the Fulbright award in 2008, a Chang-Jiang Chair Professorship of the Chinese Ministry of Education in 2009, and the Inaugural Research Award of the College of Science at Purdue University in 2013. He serves on editorial boards for a number of leading international research journals, and has authored/coauthored over a hundred fifty peer-reviewed research articles. He has coauthored a monograph published by Science Press of China, and a book published by Springer-Verlag.  His main research interests are numerical analysis, spectral methods and scientific computing with applications in computational fluid dynamics and materials science.