We shall present in this talk some fast spectral-Galerkin algorithms developed in the last few years. Their computational complexities are comparable to those of finite difference and finite element algorithms, yet they provide much more accurate results while using a significantly smaller number of unknowns. Furthermore, the most computationally intensive components in these algorithms are matrix multiplications and/or fast Fourier transforms/fast multipole methods for which existing parallel software can be readily applied.
We shall also present recent numerical results using these
fast spectral-Galerkin algorithms for various scientific and
engineering applications, including computational fluid dynamics,
atmosphere and oceanographic sciences and material sciences.
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