Abstract
A traditional idea of the Method of Fundamental Solutions is to use some external source points where the fundamental solution should be shifted to. However, the proper definition of the locations of the sources can hardly be performed in an automated way. To circumvent this difficulty, in this paper, the source points defined along the boundary, and the collocation points are shifted to the interior of the domain together with a proper modification of the boundary conditions. Thus, the problem of singularity is avoided. The modified boundary conditions are defined on the basis of the tools of the classical finite difference methods. Several schemes are presented. The schemes can be embedded in a multi-level context in a natural way. The proposed method avoids the computational difficulties due to ill-conditioned matrices and also reduces the computational complexity of the Method of Fundamental Solutions.
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References
Alves, C.J.S., Chen, C.S., Šarler, B.: The method of fundamental solutions for solving Poisson problems. Int. Ser. Adv. Bound. Elem. 13, 67–76 (2002)
Chen, C.S., Karageorghis, A., Li, Y.: On choosing the location of the sources in the MFS. Numer. Algorithms 72, 107–130 (2016)
Fam, G.S.A., Rashed, Y.F.: A study on the source points locations in the method of fundamental solutions. Int. Ser. Adv. Bound. Elem. 13, 297–312 (2002)
Gáspár, C.: Some variants of the method of fundamental solutions: regularization using radial and nearly radial basis functions. Cent. Eur. J. Math. 11(8), 1429–1440 (2013)
Gáspár, C.: A regularized multi-level technique for solving potential problems by the method of fundamental solutions. Eng. Anal. Bound. Elem. 57, 66–71 (2015)
Gáspár, C.: Fast meshless techniques based on the regularized method of fundamental solutions. In: Dimov, I., Faragó, I., Vulkov, L. (eds.) Numerical Analysis and Its Applications, NAA 2016. LNCS, vol. 10187, pp. 326–333. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-57099-0_36
Golberg, M.A.: The method of fundamental solutions for Poisson’s equation. Eng. Anal. Bound. Elem. 16, 205–213 (1995)
Gu, Y., Chen, W., Zhang, J.: Investigation on near-boundary solutions by singular boundary method. Eng. Anal. Bound. Elem. 36, 1173–1182 (2012)
Šarler, B.: Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions. Eng. Anal. Bound. Elem. 33, 1374–1382 (2009)
Young, D.L., Chen, K.H., Lee, C.W.: Novel meshless method for solving the potential problems with arbitrary domain. J. Comput. Phys. 209, 290–321 (2005)
Acknowledgments
The research was partly supported by the European Union and the Hungarian Government from the project ‘FIEK - Center for cooperation between higher education and the industries at the Széchenyi István University’ under grant number GINOP-2.3.4-15-2016-00003.
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Gáspár, C. (2019). The Method of Fundamental Solutions Combined with a Multi-level Technique. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods. Theory and Applications. FDM 2018. Lecture Notes in Computer Science(), vol 11386. Springer, Cham. https://doi.org/10.1007/978-3-030-11539-5_26
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DOI: https://doi.org/10.1007/978-3-030-11539-5_26
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