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Fast Fourier-Galerkin Methods for Nonlinear Boundary Integral Equations

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Abstract

We develop in this paper a fast Fourier-Galerkin method for solving the nonlinear integral equation which is reformulated from a class of nonlinear boundary value problems. By projecting the nonlinear term onto the approximation subspaces, we make the Fourier-Galerkin method more efficient for solving the nonlinear integral equations. A fast algorithm for solving the resulting discrete nonlinear system is designed by integrating together the techniques of matrix compressing, numerical quadrature for oscillatory integrals, and the multilevel augmentation method. We prove that the proposed method enjoys an optimal convergence order and a nearly linear computational complexity. Numerical experiments are presented to confirm the theoretical estimates and to demonstrate the efficiency and accuracy of the proposed method.

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Acknowledgments

This research is partially supported by Guangdong Provincial Government of China through the “Computational Science Innovative Research Team” program, by Guangdong Province Key Lab of Computational Science, by the Natural Science Foundation of China under Grants 11071286, 91130009, 11071250, 11126149 and 11271370. The third author is also supported in part by US Air Force Office of Scientific Research under Grant FA9550-09-1-0511 and by the US National Science Foundation under Grant DMS-1115523.

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Authors and Affiliations

  1. Department of Mathematics and System Science, College of Science, National University of Defense Technology, Changsha, 410073, People’s Republic of China

    Xiangling Chen

  2. School of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China

    Rui Wang

  3. Department of Mathematics, Syracuse University, Syracuse, NY, 13244, USA

    Yuesheng Xu

  4. Guangdong Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou, 510275, People’s Republic of China

    Yuesheng Xu

Authors
  1. Xiangling Chen
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  2. Rui Wang
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  3. Yuesheng Xu
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Correspondence to Rui Wang.

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Chen, X., Wang, R. & Xu, Y. Fast Fourier-Galerkin Methods for Nonlinear Boundary Integral Equations. J Sci Comput 56, 494–514 (2013). https://doi.org/10.1007/s10915-013-9687-y

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  • DOI: https://doi.org/10.1007/s10915-013-9687-y

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