Usually, grid discretizations of differential equations are given by
either global methods, such as spectral methods and pseudospectral
methods, or local methods, such as finite difference and finite
element methods. Global methods are highly accurate, but local
methods have much better flexibility for handling complex
geometries and boundary conditions. Recently, wavelet approaches
have begun to reveal their power as a fundamental alternative to
classic global and local methods. In this talk, I will review
the recent development of a wavelet-collocation scheme. This
wavelet scheme combines global methods' accuracy with local
methods' flexibility for solving scientific and engineering
problems. Example applications will be discussed to CFD, CEM,
computational mechanics and nonlinear dynamics.
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