A New Collocation Scheme Using Non-polynomial Basis Functions

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• 作者：Chao Zhang ; Wenjie Liu ; Li-Lian Wang
• 关键词：Generalised Birkhoff interpolation problem ; Non ; polynomial basis ; Well ; conditioned collocation methods
• 刊名：Journal of Scientific Computing
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：70
• 期：2
• 页码：793-818
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Algorithms; Computational Mathematics and Numerical Analysis; Appl.Mathematics/Computational Methods of Engineering; Theoretical, Mathematical and Computational Physics;
• 出版者：Springer US
• ISSN：1573-7691
• 卷排序：70

文摘

In this paper, we construct a set of non-polynomial basis functions from a generalised Birkhoff interpolation problem involving the operator: \({\mathscr {L}}_\lambda ={d^2}/{dx^2}-\lambda ^2 \) with constant \(\lambda .\) With a direct inverting the operator, the basis can be pre-computed in a fast and stable manner. This leads to new collocation schemes for general second-order boundary value problems with (i) the matrix corresponding to the operator \({\mathscr {L}}_\lambda \) being identity; (ii) well-conditioned linear systems and (iii) exact imposition of various boundary conditions. This also provides efficient solvers for time-dependent nonlinear problems. Moreover, we can show that the new basis has the approximability to general functions in Sobolev spaces as good as orthogonal polynomials.