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.
@article{bwmeta1.element.doi-10_2478_s11533-013-0251-7,
author = {Csaba G\'asp\'ar},
title = {Some variants of the method of fundamental solutions: regularization using radial and nearly radial basis functions},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {1429-1440},
zbl = {1273.65190},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0251-7}
}
Csaba Gáspár. Some variants of the method of fundamental solutions: regularization using radial and nearly radial basis functions. Open Mathematics, Tome 11 (2013) pp. 1429-1440. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0251-7/
[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 | Zbl 1011.65086
[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/10.1088/0305-4470/16/12/029
[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)00017-6 | Zbl 1006.65500
[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.2005.04.001 | Zbl 1182.74250
[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.2009.12.002 | Zbl 1244.65219
[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 | Zbl 1011.65092
[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 | Zbl 1118.74366
[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.2003.04.001 | Zbl 1077.65127
[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
[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_9
[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.2012.04.054
[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.2012.01.006
[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
[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
[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.2009.06.008 | Zbl 1244.76084
[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.2005.03.007 | Zbl 1073.65139