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Galerkin Formulations of the Method of Fundamental Solutions

Published online by Cambridge University Press:  03 June 2015

J.R. Berger  and
Andreas Karageorghis
J.R. Berger*
Affiliation:
Department of Mechanical Engineering, Colorado School of Mines, Golden, CO 80401, USA
Andreas Karageorghis*
Affiliation:
Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
*
Corresponding author. Email: [email protected]
Email: [email protected]
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Abstract

In this paper, we introduce two Galerkin formulations of the Method of Fundamental Solutions (MFS). In contrast to the collocation formulation of the MFS, the proposed Galerkin formulations involve the evaluation of integrals over the boundary of the domain under consideration. On the other hand, these formulations lead to some desirable properties of the stiffness matrix such as symmetry in certain cases. Several numerical examples are considered by these methods and their various features compared.

Keywords

65N3865N3065N3565N80Galerkin formulationLaplace equationmethod of fundamental solutions
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Special Issue 4 , August 2013 , pp. 423 - 441
Copyright
Copyright © Global-Science Press 2013

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References

[1]Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications Inc., New York, 1992, Reprint of the 1972 edition.Google Scholar
[2]Berger, J. and Karageorghis, A., The method of fundamental solutions for heat conduction in layered materials, Int. J. Numer. Methods Eng., 45 (1999), pp. 1681–1694.3.0.CO;2-T>CrossRefGoogle Scholar
[3]Cheung, Y. K., Jin, W. G. and Zienkiewicz, O. C., Direct solution procedure for solution of harmonic problems using complete, nonsingular, Trefftz functions, Commun. Appl. Numer. Methods, 5 (1989), pp. 159–169.CrossRefGoogle Scholar
[4]Davis, P. J., Circulant Matrices, John Wiley & Sons, New York-Chichester-Brisbane, 1979.Google Scholar
[5]Fairweather, G. and Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9 (1998), pp. 69–95.Google Scholar
[6]Fairweather, G. and Johnston, R. L., The method of fundamental solutions for problems in potential theory, In: Treatment of Integral Equations by Numerical Methods, (Baker, C. T. H. and Miller, G. F., eds.), Academic Press, London, 1982, pp. 349–359.Google Scholar
[7]Fairweather, G., Karageorghis, A. and Martin, P. A., The method of fundamental solutions for scattering and radiation problems, Eng. Anal. Bound. Elem., 27 (2003), pp. 759–769.CrossRefGoogle Scholar
[8]Golberg, M. A. and Chen, C. S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, Boundary Integral Methods: Numerical and Mathematical Aspects (Golberg, M. A., ed.), Comput. Eng., Vol. 1, WIT Press/ Comput. Mech. Publ., Boston, MA, 1999, pp. 103–176.Google Scholar
[9]Han, P. S., Olson, M. D. and Johnston, R. L., A Galerkin boundary element formulation with moving singularities, Eng. Comput., 1 (1984), pp. 232–236.CrossRefGoogle Scholar
[10]Heise, U., Application of the singularity method for the formulation of plane elastostatical boundary value problems as integral equations, Acta Mech., 31 (1978/79), pp. 33–69.Google Scholar
[11]Jin, W. G. and Cheung, Y. K., Trefftz direct method, Adv. Eng. Softw., 24 (1995), pp. 65–69.Google Scholar
[12]Jin, W. G., Sheng, N., Sze, K. Y. and Li, J., Trefftz indirect methods for plane piezoelectricity, Int. J. Numer. Methods Eng., 63 (2005), pp. 139–158.Google Scholar
[13]Johnson, C., Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover Publications, New York, 2009.Google Scholar
[14]Johnston, R. L. and Fairweather, G., The method of fundamental solutions for problems in potential flow, Appl. Math. Model., 8 (1984), pp. 265–270.Google Scholar
[15]Kita, E., and Kamiya, N., Trefftz method: an overview, Adv. Eng. Softw., 24 (1995), pp. 3–12.CrossRefGoogle Scholar
[16]Limić, N., Galerkin-Petrov method for Helmholtz equation exterior problems, Glas. Mat. Ser. III, 16 (1981), pp. 245–260.Google Scholar
[17]Liu, Y. J., Mukherjee, S., Nishimura, N., Schanz, M., Ye, W., Sutradhar, A., Pan, E., Dumont, N. A., Frangi, A. and Saez, A., Recent advances and emerging applications of the boundary element method, Appl. Mech. Rev., 64 (2011), 031001.Google Scholar
[18]Mathon, R. and Johnston, R. L., The approximate solution of elliptic boundary–value problems by fundamental solutions, SIAM J. Numer. Anal., 14 (1977), pp. 638–650.CrossRefGoogle Scholar
[19]Portela, A. and Charafi, A., Treftz boundary element method for domains with slits, Eng. Anal. Bound. Elem., 20 (1997), pp. 299–304.Google Scholar
[20]Portela, A. and Charafi, A., Treftz boundary elements–multi-region formulations, Int. J. Numer. Methods Eng., 45 (1999), pp. 821–840.3.0.CO;2-S>CrossRefGoogle Scholar
[21]Smyrlis, Y.-S. and Karageorghis, A., Some aspects of the method of fundamental solutions for certain harmonic problems, J. Sci. Comput., 16 (2001), pp. 341–371.Google Scholar
[22]Sutradhar, A., Paulino, G. H. and Gray, L. J., The Symmetric Galerkin Boundary Element Method, Springer-Verlag, Berlin, 2008.Google Scholar
[23]Trefftz, E., Ein Gegenstuück zum Ritz schen Verfahren, Proceedings, 2nd International Congress on Applied Mechanics, Zurich, 1926, pp. 131–137.Google Scholar