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Multilevel augmentation methods for differential equations

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Abstract

We develop multilevel augmentation methods for solving differential equations. We first establish a theoretical framework for convergence analysis of the boundary value problems of differential equations, and then construct multiscale orthonormal bases in H m0 (0,1) spaces. Finally, the multilevel augmentation methods in conjunction with the multiscale orthonormal bases are applied to two-point boundary value problems of both second-order and fourth-order differential equations. Theoretical analysis and numerical tests show that these methods are computationally stable, efficient and accurate.

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Author information

Authors and Affiliations

  1. Department of Scientific Computing and Computer Applications, Zhongshan University, Guangzhou, 510275, P. R. China

    Zhongying Chen & Bin Wu

  2. Department of Mathematics, Syracuse University, Syracuse, New York 13244-1150, USA and Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100080, P. R. China

    Yuesheng Xu

Authors
  1. Zhongying Chen
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  2. Bin Wu
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  3. Yuesheng Xu
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Corresponding author

Correspondence to Zhongying Chen.

Additional information

Communicated by J. Xu

Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday with friendship and esteem.

Mathematics subject classifications (2000)

65J15, 65R20.

Zhongying Chen: Supported in part by the Natural Science Foundation of China under grants 10371137 and 10201034, the Foundation of Doctoral Program of National Higher Education of China under grant 20030558008, Guangdong Provincial Natural Science Foundation of China under grant 1011170 and the Foundation of Zhongshan University Advanced Research Center.

Yuesheng Xu: Corresponding author. Supported in part by the US National Science Foundation under grants 9973427 and 0312113, by NASA under grant NCC5-399, by the Natural Science Foundation of China under grant 10371122 and by the Chinese Academy of Sciences under the program of “One Hundred Distinguished Young Scientists”.

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Chen, Z., Wu, B. & Xu, Y. Multilevel augmentation methods for differential equations. Adv Comput Math 24, 213–238 (2006). https://doi.org/10.1007/s10444-004-4092-6

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  • DOI: https://doi.org/10.1007/s10444-004-4092-6

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