Fast Spectral-Galerkin Methods and their Applications

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As computer simulations are playing an ever increasing role in many branches of science and engineering and are rapidly replacing much of the expensive prototyping and testing phases in manufacturing and in science explorations, fast and robust numerical methods are becoming an indispensable tool for many scientists and engineers. The focus of this project is to design auurate, fast and robust spectral methods for solving a large class of partial differential equations, and apply them to investigate several important problems of current interest. The proposed research will result in fast and accurate numerical algorithms for a class of partial differential equations with applications in acoustic and electromagnetic scattering, complex fluids and materials science. In particular, the proposed numerical algorithms will make it possible to directly solve some important high-dimensional equations which are currently not treatable with existing numerical algorithms.It is expected that the proposed numerical simulations will enable us to handle challenging problems having stringent accuracy and/or memory requirements with a reasonable cost in CPU and turn-around time, contribute towards better understandings of the complex physical and mathematical problems, and provide valuable information for the design of advanced materials and on the rheological and hydrodynamic properties of complex fluids. Another important goal of this project is to engage graduate and undergraduate students in learning necessary skills of computational and applied mathematics so that they can pursue a successful career in sciences and engineering.