Fast Meshless Techniques Based on the Regularized Method of Fundamental Solutions

  • Csaba GáspárEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10187)


A regularized method of fundamental solutions is presented. The method can handle Neumann and mixed boundary conditions as well without using a desingularization technique. Instead, the approach combines the regularized method of fundamental solutions with traditional finite difference techniques based on some inner collocation points located in the vicinity of the boundary collocation points. Nevertheless, the resulting method remains meshless. The method avoids the problem of singularity and can be embedded in a natural multi-level context. The method is illustrated via a numerical example.


Meshless methods Method of fundamental solutions Regularization Finite difference schemes Multi-level methods 


  1. 1.
    Chen, W., Shen, L.J., Shen, Z.J., Yuan, G.W.: Boundary knot method for Poisson equations. Eng. Anal. Boundary Elem. 29, 756–760 (2005)CrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, W., Wang, F.Z.: A method of fundamental solutions without fictitious boundary. Eng. Anal. Boundary Elem. 34, 530–532 (2010)CrossRefzbMATHGoogle Scholar
  3. 3.
    Gáspár, C.: Some variants of the method of fundamental solutions: regularization using radial and nearly radial basis functions. Central Eur. J. Math. 11(8), 1429–1440 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Gáspár, C.: A regularized multi-level technique for solving potential problems by the method of fundamental solutions. Eng. Anal. Boundary Elem. 57, 66–71 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Golberg, M.A.: The method of fundamental solutions for Poisson’s equation. Eng. Anal. Boundary Elem. 16, 205–213 (1995)CrossRefGoogle Scholar
  6. 6.
    Gu, Y., Chen, W., Zhang, J.: Investigation on near-boundary solutions by singular boundary method. Eng. Anal. Boundary Elem. 36, 1173–1182 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Li, X.: On convergence of the method of fundamental solutions for solving the Dirichlet problem of Poisson’s equation. Adv. Comput. Math. 23, 265–277 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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. Boundary Elem. 33, 1374–1382 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Stüben, K., Trottenberg, U.: Multigrid methods: fundamental algorithms, model problem analysis and applications. In: GDM-Studien, vol. 96, Birlinghoven, Germany (1984)Google Scholar
  10. 10.
    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)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Széchenyi István UniversityGyörHungary

Personalised recommendations