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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access May 22, 2013

Some variants of the method of fundamental solutions: regularization using radial and nearly radial basis functions

Csaba Gáspár
From the journal Open Mathematics

Abstract

The method of fundamental solutions and some versions applied to mixed boundary value problems are considered. Several strategies are outlined to avoid the problems due to the singularity of the fundamental solutions: the use of higher order fundamental solutions, and the use of nearly fundamental solutions and special fundamental solutions concentrated on lines instead of points. The errors of the approximations as well as the problem of ill-conditioned matrices are illustrated via numerical examples.

MSC: 65N80; 65N99

[1] Alves C.J.S., Chen C.S., Šarler B., The method of fundamental solutions for solving Poisson problems, In: Boundary Elements XXIV, Sintra, June, 2002, Adv. Bound. Elem., 13, WIT Press, Southampton, 2002, 67–76 Search in Google Scholar

[2] Atkinson W.J., Young J.H., Brezovich I.A., An analytic solution for the potential due to a circular parallel plate capacitor, J. Phys. A, 1983, 16(12), 2837–2841 http://dx.doi.org/10.1088/0305-4470/16/12/02910.1088/0305-4470/16/12/029Search in Google Scholar

[3] Chen W., Symmetric boundary knot method, Eng. Anal. Bound. Elem., 2002, 26(6), 489–494 http://dx.doi.org/10.1016/S0955-7997(02)00017-610.1016/S0955-7997(02)00017-6Search in Google Scholar

[4] Chen W., Shen L.J., Shen Z.J., Yuan G.W., Boundary knot method for Poisson equations, Eng. Anal. Bound. Elem., 2005, 29(8), 756–760 http://dx.doi.org/10.1016/j.enganabound.2005.04.00110.1016/j.enganabound.2005.04.001Search in Google Scholar

[5] Chen W., Wang F.Z., A method of fundamental solutions without fictitious boundary, Eng. Anal. Bound. Elem., 2010, 34(5), 530–532 http://dx.doi.org/10.1016/j.enganabound.2009.12.00210.1016/j.enganabound.2009.12.002Search in Google Scholar

[6] Fam G.S.A., Rashed Y.F., A study on the source points locations in the method of fundamental solutions, In: Boundary Elements XXIV, Sintra, June, 2002, Adv. Bound. Elem., 13, WIT Press, Southampton, 2002, 297–312 Search in Google Scholar

[7] Fam G.S.A., Rashed Y.F., The method of fundamental solutions, a dipole formulation for potential problems, In: Boundary Elements XXVI, Bologna, April 19–21, 2004, Adv. Bound. Elem., 19, WIT Press, Southampton, 2004, 193–203 Search in Google Scholar

[8] Gáspár C., A meshless polyharmonic-type boundary interpolation method for solving boundary integral equations, Eng. Anal. Bound. Elem., 2004, 28(10), 1207–1216 http://dx.doi.org/10.1016/j.enganabound.2003.04.00110.1016/j.enganabound.2003.04.001Search in Google Scholar

[9] Gáspár C., A multi-level regularized version of the method of fundamental solutions, In: The Method of Fundamental Solutions — A Meshless Method, Dynamic, Atlanta, 2008, 145–164 Search in Google Scholar

[10] Gáspár C., Several meshless solution techniques for the Stokes flow equations, In: Progress on Meshless Methods, Comput. Methods Appl. Sci., 11, Springer, New York, 2009, 141–158 http://dx.doi.org/10.1007/978-1-4020-8821-6_910.1007/978-1-4020-8821-6_9Search in Google Scholar

[11] Gu Y., Chen W., He X.-Q., Singular boundary method for steady-state heat conduction in three dimensional general anisotropic media, International Journal of Heat and Mass Transfer, 2012, 55(17–18), 4837–4848 http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.04.05410.1016/j.ijheatmasstransfer.2012.04.054Search in Google Scholar

[12] Gu Y., Chen W., Zhang J., Investigation on near-boundary solutions by singular boundary method, Eng. Anal. Bound. Elem., 2012, 36(8), 1173–1182 http://dx.doi.org/10.1016/j.enganabound.2012.01.00610.1016/j.enganabound.2012.01.006Search in Google Scholar

[13] Šarler B., Desingularised method of double layer fundamental solutions for potential flow problems, In: Boundary Elements and Other Mesh Reduction Methods XXX, Maribor, July 7–9, 2008, WIT Trans. Model. Simul., 47, WIT Press, Southampton, 2008, 159–168 10.2495/BE080161Search in Google Scholar

[14] Šarler B., A modified method of fundamental solutions for potential flow problems, In: The Method of Fundamental Solutions — A Meshless Method, Dynamic, Atlanta, 2008, 299–326 10.2495/BE080161Search in Google Scholar

[15] 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., 2009, 33(12), 1374–1382 http://dx.doi.org/10.1016/j.enganabound.2009.06.00810.1016/j.enganabound.2009.06.008Search in Google Scholar

[16] Young D.L., Chen K.H., Lee C.W., Novel meshless method for solving the potential problems with arbitrary domain, J. Comput. Phys., 2005, 209(1), 290–321 http://dx.doi.org/10.1016/j.jcp.2005.03.00710.1016/j.jcp.2005.03.007Search in Google Scholar

Published Online: 2013-5-22
Published in Print: 2013-8-1

© 2013 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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