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The Method of Fundamental Solutions: A Weighted Least-Squares Approach

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Abstract

We investigate the Method of Fundamental Solutions (MFS) for the solution of certain elliptic boundary value problems. In particular, we study the case in which the number of collocation points exceeds the number of singularities, which leads to an over-determined linear system. In such a case, the resulting linear system is over-determined and the proposed algorithm chooses the approximate solution for which the error, when restricted to the boundary, minimizes a suitably defined discrete Sobolev norm. This is equivalent to a weighted least-squares treatment of the resulting over-determined system. We prove convergence of the method in the case of the Laplace’s equation with Dirichlet boundary data in the disk. We develop an alternative way of implementing the numerical algorithm, which avoids the inherent ill-conditioning of the MFS matrices. Finally, we present numerical experiments suggesting that introduction of Sobolev weights improves the approximation.

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References

  1. R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edn., Pure and Applied Mathematics, vol. 140, Academic Press, Amsterdam, 2003.

  2. A. Bogomolny, Fundamental solutions method for elliptic boundary value problems, SIAM J. Numer. Anal., 22 (1985), pp. 644–669.

  3. P. J. Davis, Circulant Matrices, A Wiley–Interscience Publication, Pure and Applied Mathematics, John Wiley & Sons, New York–Chichester–Brisbane, 1979.

  4. G. Fairweather and A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems. Numerical treatment of boundary integral equations, Adv. Comput. Math., 9 (1998), pp. 69–95.

    Google Scholar 

  5. G. Fairweather, A. Karageorghis, and P. A. Martin, The method of fundamental solutions for scattering and radiation problems, Eng. Anal. Bound. Elem., 27 (2003), pp. 759–769.

  6. M. A. Golberg and C. S. Chen, Discrete Projection Methods for Integral Equations, Computational Mechanics Publications, Southampton, 1997.

  7. M. A. Golberg and C. S. Chen, The method of fundamental solutions for potential, Helmholtz and diffusion problems, Boundary Integral Methods: Numerical and Mathematical Aspects, Comput. Eng., vol. 1, WIT Press/Comput. Mech. Publ., Boston, MA, 1999, pp. 103–176.

  8. M. Katsurada, A mathematical study of the charge simulation method. II, J. Fac. Sci., Tokyo Univ., Sect. IA Math., 36 (1989), pp. 135–162.

  9. M. Katsurada and H. Okamoto, A mathematical study of the charge simulation method. I, J. Fac. Sci., Tokyo Univ., Sect. IA Math., 35 (1988), pp. 507–518.

  10. T. Kitagawa, On the numerical stability of the method of fundamental solution applied to the Dirichlet problem, Japan J. Appl. Math., 5 (1988), pp. 123–133.

    Google Scholar 

  11. T. Kitagawa, Asymptotic stability of the fundamental solution method, Proceedings of the International Symposium on Computational Mathematics (Matsuyama, 1990), J. Comput. Appl. Math., 38 (1991), pp. 263–269.

  12. J. A. Kołodziej, Review of applications of the boundary collocation methods in mechanics of continuous media, Solid Mech. Arch., 12 (1987), pp. 187–231.

  13. J. A. Kołodziej, Applications of the Boundary Collocation Method in Applied Mechanics, (in Polish), Wydawnictwo Politechniki Poznańskiej, Poznań, 2001.

  14. V. D. Kupradze and M. A. Aleksidze, An approximate method of solving certain boundary-value problems (in Russian), Soobšč. Akad. Nauk Gruzin. SSR, 30 (1963), pp. 529–536.

  15. V. D. Kupradze and M. A. Aleksidze, The method of functional equations for the approximate solution of certain boundary-value problems (in Russian), Ž. Vyčisl. Mat. Mat. Fiz., 4 (1964), pp. 683–715.

  16. M. Maiti and S. K. Chakrabarty, Integral equation solutions for simply supported polygonal plates, Int. J. Eng. Sci., 12 (1974), pp. 793–806.

    Google Scholar 

  17. R. Mathon and R. L. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J. Numer. Anal., 14 (1977), pp. 638–650.

    Google Scholar 

  18. P. A. Ramachandran, Method of fundamental solutions: Singular value decomposition analysis, Commun. Numer. Methods Eng., 18 (2002), pp. 789–801.

  19. I. Saavedra and H. Power, Multipole fast algorithm for the least-squares approach of the method of fundamental solutions for three-dimensional harmonic problems, Numer. Methods Partial Differ. Equations, 19 (2003), pp. 828–845.

  20. Y.-S. Smyrlis and A. Karageorghis, Some aspects of the method of fundamental solutions for certain harmonic problems, J. Sci. Comput., 16 (2001), pp. 341–371.

    Google Scholar 

  21. Y.-S. Smyrlis and A. Karageorghis, A linear least-squares MFS for certain elliptic problems, Numer. Algorithms, 35 (2004), pp. 29–44.

    Google Scholar 

  22. Y.-S. Smyrlis and A. Karageorghis, Numerical analysis of the MFS for certain harmonic problems, M2AN Math. Model. Numer. Anal., 38 (2004), pp. 495–517.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678, Nicosia, Cyprus

    Yiorgos-Sokratis Smyrlis

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  1. Yiorgos-Sokratis Smyrlis
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Correspondence to Yiorgos-Sokratis Smyrlis.

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AMS subject classification (2000)

35E05, 35J25, 65N12, 65N15, 65N35, 65T50

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Smyrlis, YS. The Method of Fundamental Solutions: A Weighted Least-Squares Approach. Bit Numer Math 46, 163–194 (2006). https://doi.org/10.1007/s10543-006-0043-6

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  • DOI: https://doi.org/10.1007/s10543-006-0043-6

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