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Applicability and applications of the method of fundamental solutions


Author: Yiorgos-Sokratis Smyrlis
Journal: Math. Comp. 78 (2009), 1399-1434
MSC (2000): Primary 35E05, 41A30, 65N35; Secondary 35G15, 35J40, 65N38
DOI: https://doi.org/10.1090/S0025-5718-09-02191-7
Published electronically: January 30, 2009
MathSciNet review: 2501056
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Abstract: In the present work, we investigate the applicability of the method of fundamental solutions for the solution of boundary value problems of elliptic partial differential equations and elliptic systems. More specifically, we study whether linear combinations of fundamental solutions can approximate the solutions of the boundary value problems under consideration. In our study, the singularities of the fundamental solutions lie on a prescribed pseudo–boundary — the boundary of a domain which embraces the domain of the problem under consideration. We extend previous density results of Kupradze and Aleksidze, and of Bogomolny, to more general domains and partial differential operators, and with respect to more appropriate norms. Our domains may possess holes and their boundaries are only required to satisfy a rather weak boundary requirement, namely the segment condition. Our density results are with respect to the norms of the spaces $C^\ell (\overline {\Omega })$. Analogous density results are obtainable with respect to Hölder norms. We have studied approximation by fundamental solutions of the Laplacian, biharmonic and $m-$harmonic and modified Helmholtz and poly–Helmholtz operators. In the case of elliptic systems, we obtain analogous density results for the Cauchy–Navier operator as well as for an operator which arises in the linear theory of thermo–elasticity. We also study alternative formulations of the method of fundamental solutions in cases when linear combinations of fundamental solutions of the equations under consideration are not dense in the solution space. Finally, we show that linear combinations of fundamental solutions of operators of order $m\ge 4$, with singularities lying on a prescribed pseudo–boundary, are not in general dense in the corresponding solution space.


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  • Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
  • Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
  • M. A. Aleksidze, On the question of a practical application of a new approximation method, Differencial′nye Uravnenija 2 (1966), 1625–1629 (Russian). MR 0203975
  • M. A. Aleksidze, Fundamental′nye funktsii v priblizhennykh resheniyakh granichnykh zadach, Spravochnaya Matematicheskaya Biblioteka. [Mathematical Reference Library], “Nauka”, Moscow, 1991 (Russian). With an English summary. MR 1154556
  • E. Almansi, Sull’integrazione dell’equazione differenziale $\Delta ^{2}=0$, Atti. Reale. Accad. Sci. Torino 31 (1896), 881–888.
  • ---, Sull’integrazione dell’equazione differenziale $\Delta ^{2n}=0$, Annali di Mathematica Pura et Applicata, Series III 2 (1898), 1–51.
  • C. J. S. Alves, Inverse scattering with spherical incident waves, Mathematical and numerical aspects of wave propagation (Golden, CO, 1998), SIAM, Philadelphia, PA, 1998, pp. 502–504.
  • Nachman Aronszajn, Thomas M. Creese, and Leonard J. Lipkin, Polyharmonic functions, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1983. Notes taken by Eberhard Gerlach; Oxford Science Publications. MR 745128
  • Alexander Bogomolny, Fundamental solutions method for elliptic boundary value problems, SIAM J. Numer. Anal. 22 (1985), no. 4, 644–669. MR 795946, DOI https://doi.org/10.1137/0722040
  • Felix E. Browder, Approximation by solutions of partial differential equations, Amer. J. Math. 84 (1962), 134–160. MR 178247, DOI https://doi.org/10.2307/2372809
  • C. S. Chen, M. Ganesh, M. A. Golberg, and A. H.-D. Cheng, Multilevel compact radial functions based computational schemes for some elliptic problems, Comput. Math. Appl. 43 (2002), no. 3-5, 359–378. Radial basis functions and partial differential equations. MR 1883573, DOI https://doi.org/10.1016/S0898-1221%2801%2900292-9
  • A. H.-D. Cheng, H. Antes, and N. Ortner, Fundamental solutions of products of Helmholtz and polyharmonic operators, Eng. Anal. Bound. Elem. 14 (1994), no. 2, 187–191.
  • H. A. Cho, M. A. Golberg, A. S. Muleshkov, and X. Li, Trefftz methods for time dependent partial differential equations, Comput. Mat. Cont. 1 (2004), no. 1, 1–37.
  • A. Doicu, Y. Eremin, and T. Wriedt, Acoustic and Electromagnetic Scattering Analysis using Discrete Sources, Academic Press, New York, 2000.
  • Leon Ehrenpreis, On the theory of kernels of Schwartz, Proc. Amer. Math. Soc. 7 (1956), 713–718. MR 82637, DOI https://doi.org/10.1090/S0002-9939-1956-0082637-9
  • Graeme Fairweather and Andreas Karageorghis, The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math. 9 (1998), no. 1-2, 69–95. Numerical treatment of boundary integral equations. MR 1662760, DOI https://doi.org/10.1023/A%3A1018981221740
  • G. Fairweather, A. Karageorghis, and P. A. Martin, The method of fundamental solutions for scattering and radiation problems, Eng. Anal. Bound. Elem. 27 (2003), 759–769.
  • Graeme Fairweather, Andreas Karageorghis, and Yiorgos-Sokratis Smyrlis, A matrix decomposition MFS algorithm for axisymmetric biharmonic problems, Adv. Comput. Math. 23 (2005), no. 1-2, 55–71. MR 2131993, DOI https://doi.org/10.1007/s10444-004-1808-6
  • Gerald B. Folland, Real analysis, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999. Modern techniques and their applications; A Wiley-Interscience Publication. MR 1681462
  • David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
  • M. A. Golberg and C. S. Chen, Discrete projection methods for integral equations, Computational Mechanics Publications, Southampton, 1997. MR 1445293
  • M. A. Golberg and C. S. Chen, The method of fundamental solutions for potential, Helmholtz and diffusion problems, Boundary integral methods: numerical and mathematical aspects, Comput. Eng., vol. 1, WIT Press/Comput. Mech. Publ., Boston, MA, 1999, pp. 103–176. MR 1690853, DOI https://doi.org/10.1007/BF01200068
  • I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 6th ed., Academic Press, Inc., San Diego, CA, 2000. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. MR 1773820
  • Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035
  • Andreas Karageorghis and Graeme Fairweather, The method of fundamental solutions for the numerical solution of the biharmonic equation, J. Comput. Phys. 69 (1987), no. 2, 434–459. MR 888063, DOI https://doi.org/10.1016/0021-9991%2887%2990176-8
  • ---, The Almansi method of fundamental–solutions for solving biharmonic problems, Int. J. Numer. Meth. Engng. 26 (1988), no. 7, 1665–1682.
  • Masashi Katsurada, A mathematical study of the charge simulation method. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), no. 1, 135–162. MR 991024
  • Masashi Katsurada, Asymptotic error analysis of the charge simulation method in a Jordan region with an analytic boundary, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37 (1990), no. 3, 635–657. MR 1080874
  • Takashi Kitagawa, On the numerical stability of the method of fundamental solution applied to the Dirichlet problem, Japan J. Appl. Math. 5 (1988), no. 1, 123–133. MR 924746, DOI https://doi.org/10.1007/BF03167903
  • J. A. Kołodziej, Zastosowanie metody kollokacji brzegowej w zagadnieniach mechaniki. (Polish) [Applications of the Boundary Collocation Method in Applied Mechanics], Wydawnictwo Politechniki Poznańskiej, Poznań, 2001.
  • V. D. Kupradze, On a method of solving approximately the limiting problems of mathematical physics, Ž. Vyčisl. Mat i Mat. Fiz. 4 (1964), 1118–1121 (Russian). MR 174183
  • V. D. Kupradze, Potential methods in the theory of elasticity, Israel Program for Scientific Translations, Jerusalem; Daniel Davey & Co., Inc., New York, 1965. Translated from the Russian by H. Gutfreund; Translation edited by I. Meroz. MR 0223128
  • V. D. Kupradze and M. A. Aleksidze, An approximate method of solving certain boundary-value problems, Soobšč. Akad. Nauk Gruzin. SSR 30 (1963), 529–536 (Russian). MR 0155098
  • V. D. Kupradze, T. G. Gegelia, M. O. Basheleshvili, and T. V. Burchuladze, Trekhmernye zadachi matematicheskoĭ teorii uprugosti i termouprugosti. (Russian) [Three–dimensional problems in the mathematical theory of elasticity and thermoelasticity.], Izdat. “Nauka”, Moscow, 1976.
  • Prem K. Kythe, Fundamental solutions for differential operators and applications, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1395370
  • A. E. H. Love, A treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944. Fourth Ed. MR 0010851
  • M. Maiti and S. K. Chakrabarty, Integral equation solutions for simply supported polygonal plates, Int. J. Engng. Sci. 12 (1974), no. 10, 793–806.
  • Bernard Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 271–355 (French). MR 86990
  • Rudolf Mathon and R. L. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J. Numer. Anal. 14 (1977), no. 4, 638–650. MR 520337, DOI https://doi.org/10.1137/0714043
  • S. N. Mergelyan, Uniform approximations of functions of a complex variable, Uspehi Matem. Nauk (N.S.) 7 (1952), no. 2(48), 31–122 (Russian). MR 0051921
  • M. Nicolescu, Les fonctions polyharmoniques, Hermann, Paris, 1936.
  • Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0365062
  • C. Runge, Zur Theorie der Eindeutigen Analytischen Functionen, Acta Math. 6 (1885), no. 1, 229–244 (German). MR 1554664, DOI https://doi.org/10.1007/BF02400416
  • L. Schwartz, Théorie des distributions. Tome I, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1091, Hermann & Cie., Paris, 1950 (French). Publ. Inst. Math. Univ. Strasbourg 9. MR 0035918
  • Yiorgos-Sokratis Smyrlis, The method of fundamental solutions: a weighted least-squares approach, BIT 46 (2006), no. 1, 163–194. MR 2214854, DOI https://doi.org/10.1007/s10543-006-0043-6
  • Y.-S. Smyrlis, Mathematical foundation of the MFS for certain elliptic systems in linear elasticity, Numer. Math. (2009), DOI: 10.1007/s00211-008-0207.1.
  • ---, Approximations by solutions of elliptic equations in semilocal spaces, J. Math. Anal. Appl. 350 (2009), no. 1, 122–134.
  • Y.-S. Smyrlis and A. Karageorghis, The under–determined version of the MFS: Taking more sources than collocation points, Submitted for publication.
  • Yiorgos-Sokratis Smyrlis and Andreas Karageorghis, A linear least-squares MFS for certain elliptic problems, Numer. Algorithms 35 (2004), no. 1, 29–44. MR 2041801, DOI https://doi.org/10.1023/B%3ANUMA.0000016581.85429.8d
  • Yiorgos-Sokratis Smyrlis and Andreas Karageorghis, Numerical analysis of the MFS for certain harmonic problems, M2AN Math. Model. Numer. Anal. 38 (2004), no. 3, 495–517. MR 2075757, DOI https://doi.org/10.1051/m2an%3A2004023
  • P. Sundqvist, Numerical Computations with Fundamental Solutions (Numeriska beräkingar med fundamentallösningar), Ph.D. thesis, University of Uppsala, Faculty of Science and Technology, May 2005.
  • Nikolai N. Tarkhanov, The Cauchy problem for solutions of elliptic equations, Mathematical Topics, vol. 7, Akademie Verlag, Berlin, 1995. MR 1334094
  • E. Trefftz, Ein Gegenstück zum Ritzschen Verfahren, $2^{\mathrm {er}}$ Intern. Kongr. für Techn. Mech., Zürich, 1926, pp. 131–137.
  • François Trèves, Linear partial differential equations with constant coefficients: Existence, approximation and regularity of solutions, Mathematics and its Applications, Vol. 6, Gordon and Breach Science Publishers, New York-London-Paris, 1966. MR 0224958
  • Th. Tsangaris, Y.-S. Smyrlis, and A. Karageorghis, Numerical analysis of the method of fundamental solutions for harmonic problems in annular domains, Numer. Methods Partial Differential Equations 22 (2006), no. 3, 507–539. MR 2212224, DOI https://doi.org/10.1002/num.20104
  • Teruo Ushijima and Fumihiro Chiba, Error estimates for a fundamental solution method applied to reduced wave problems in a domain exterior to a disc, Proceedings of the 6th Japan-China Joint Seminar on Numerical Mathematics (Tsukuba, 2002), 2003, pp. 137–148. MR 2022324, DOI https://doi.org/10.1016/S0377-0427%2803%2900559-4
  • Barnet M. Weinstock, Uniform approximation by solutions of elliptic equations, Proc. Amer. Math. Soc. 41 (1973), 513–517. MR 340794, DOI https://doi.org/10.1090/S0002-9939-1973-0340794-0

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Additional Information

Yiorgos-Sokratis Smyrlis
Affiliation: Department of Mathematics and Statistics, University of Cyprus, (\textgreek{Panepist’hmio K’uprou}), P. O. Box 20537, 1678 Nicosia, (\textgreek{Leukws’ia}), Cyprus, (\textgreek{K’uproc})
Email: smyrlis@ucy.ac.cy

Keywords: Trefftz methods, method of fundamental solutions, fundamental solutions, elliptic boundary value problems, approximation by special function
Received by editor(s): December 18, 2007
Published electronically: January 30, 2009
Additional Notes: This work was supported by University of Cyprus grant $\#$8037-3/312-21005.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.