Fast Spectral-Galerkin Methods and their Applications

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As computer simulations are playing an ever increasing role in manybranches of science and engineering and are rapidly replacing much ofthe expensive prototyping and testing phases in manufacturing and inscience explorations, fast and robust numerical methods are becomingan indispensable tool for many scientists and engineers. The focus ofthis project is to design fast and robust spectral-Galerkin methodsfor solving a large class of partial differential equations, and applythem to investigate several important problems of current interest.The proposed research will result in fast and accurate numericalalgorithms for a class of partial differential equations withapplications in acoustic and electromagnetic scattering, fluiddynamics and materials science. It is expected that the proposed numerical algorithms will allowsimulations of three-dimensional time-dependent problems within areasonable turn-over time, and provide capabilities to numericallyattack challenging problems arising from emerging engineering andscientific applications. In particular, the proposed numericalsimulations will contribute towards better understandings of thecomplex physical and mathematical problems, and provide valuableinformation for the design of advanced materials and on therheological and hydrodynamic properties of complex fluids. Theproposed research will also generate opportunities for undergraduateresearch projects such as software development, parallel computationand numerical simulation.