ADAPTIVE APPROXIMATION OF SOLUTIONS TO PROBLEMS WITH MULTIPLE LAYERS BY CHEBYSHEV PSEUDOSPECTRAL METHODS

We develop and analyze a family of mappings which enhance the accuracy of Chebyshev pseudo-spectral methods in approximating functions with multiple regions of localized rapid variation (layers). The mapping fa mily depends on 3N - 1 free parameters, where N is the number of layer s. N parameters depend upon the locations of the layers and on the wid ths of the layers, while the other N - 1 parameters depend on the reso lution of each layer relative to the first layer. The parameters can b e determined adaptively by minimizing a functional which measures the error of the approximation. Techniques to simplify the minimization pr ocess are developed. We further demonstrate that the appropriate choic e of mappings can lead to a significant reduction in the condition num ber of matrices associated with Chebyshev pseudo-spectral differentati on. We illustrate the effectiveness of the proposed mapping and adapti ve procedure by examples in which we approximate (i) given functions e xhibiting multiple layers and (ii) the solution of a system of partial differential equations modeling combustion in counterflowing jets so that two distinct flames occur. (C) 1995 Academic Press, Inc.