Mathematical Sciences: Fast Spectral-Galerkin Algorithms for Elliptic Problems and Efficient Solution Techniques for Unsteady Navier-Stokes Equations

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Shen 9623020 The investigator develops fast algorithms and robust software for elliptic problems and constructs, analyzes, and implements fast and accurate schemes for numerical simulation of unsteady incompressible flows. The numerical simulation of incompressible flows, which plays an important role in numerous scientific and industrial applications of current interest, is a challenging task for both numerical analysts and computational dynamicists. The proposed fast algorithms are based on the spectral discretization for the space variables and high-order projection schemes for the time variable. The proposed algorithms allow accurate numerical simulation of incompressible flows with significantly less computational effort than the traditional lower-order methods. Fast and accurate numerical solution of partial differential equations is of major importance in many fields of science and engineering, including among others high performance computing, turbulence modeling and numerical weather prediction. A main objective of this project is to develop fast algorithms with optimal computational complexity for a large class of elliptic equations, and to implement them in a high performance software package that is made available to the public through anonymous ftp and World Wide Web. It is expected that the proposed fast algorithms and software will become an important tool for many scientists and engineers.