Adaptive approximation of solutions to problems with multiple layers by Chebyshev pseudo-spectral methods
The objective of this paper is to describe an extension of the family of mappings which enhance the accuracy of Chebyshev pseudo-spectral methods in approximating functions with multiple layers. The mapping family depends on \(3N- 1\) free parameters, where \(N\) denotes the number of layers. \(N\) parameters are related to the locations of layers, \(N\) parameters depend on the width of the layers, while the other \(N- 1\) parameters depend on the resolution of each layer relative to the first layer. All parameters can be determined adaptively by minimizing the error of the approximation. The authors propose some techniques to simplify the minimization problem, and demonstrate that the appropriate choice of mappings can lead to a significant reduction in the condition number of associated differentiation matrices. In my opinion the paper is very interesting and should be mentioned by everyone involved in research of the solutions of partial differential equations occurring in many areas of applications, such as fluid dynamics, solid mechanics, combustion, solidification, and wave propagation.
scientific article; zbMATH DE number 740618