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Conformal mapping for the efficient MFS solution of Dirichlet boundary value problems

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Abstract

In this work, we use conformal mapping to transform harmonic Dirichlet problems of Laplace’s equation which are defined in simply-connected domains into harmonic Dirichlet problems that are defined in the unit disk. We then solve the resulting harmonic Dirichlet problems efficiently using the method of fundamental solutions (MFS) in conjunction with fast fourier transforms (FFTs). This technique is extended to harmonic Dirichlet problems in doubly-connected domains which are now mapped onto annular domains. The solution of the resulting harmonic Dirichlet problems can be carried out equally efficiently using the MFS with FFTs. Several numerical examples are presented.

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References

  1. Abramowitz M, Stegun IE (1972) Handbook of mathematical functions. Dover, New York

    MATH  Google Scholar 

  2. Bogomolny A (1985) Fundamental solutions method for elliptic boundary value problems. SIAM J Numer Anal 22: 644–669

    Article  MATH  MathSciNet  Google Scholar 

  3. Cho HA, Golberg MA, Muleshkov AS, Li X (2004) Trefftz methods for time dependent partial differential equations. (CMC) Comput Mater Contin 1: 1–37

    Google Scholar 

  4. Davis PJ (1979) Circulant matrices. Wiley, New York

    MATH  Google Scholar 

  5. Fairweather G, Karageorghis A (1998) The method of fundamental solutions for elliptic boundary value problems. Adv Comput Math 9: 69–95

    Article  MathSciNet  Google Scholar 

  6. Golberg MA, Chen CS (1999) The method of fundamental solutions for potential, Helmholtz and diffusion problems. In: Golberg MA (ed) Boundary integral methods and mathematical aspects. WIT Press/Computational Mechanics Publications, Boston, pp 103–176

  7. Gorzelańczyk P, Kołodziej JA (2008) Some remarks concerning the shape of the source contour with application of the method of fundamental solutions to elastic torsion of prismatic rods. Eng Anal Bound Elem 32: 64–75

    Article  Google Scholar 

  8. Hough DM (1990) User’s Guide to CONFPACK, IPS Research Report 90-11. ETH, Zürich

  9. Katsurada M (1989) A mathematical study of the charge simulation method II. J Fac Sci Univ Tokyo Sect 1A Math 36: 135–162

    MathSciNet  Google Scholar 

  10. Katsurada M, Okamoto H (1988) A mathematical study of the charge simulation method I. J Fac Sci Univ Tokyo Sect 1A Math 35: 507–518

    MathSciNet  Google Scholar 

  11. Kita E, Kamiya N (1995) Trefftz method: an overview. Adv Eng Softw 24: 3–12

    Article  Google Scholar 

  12. Kober H (1944–1948) Dictionary of conformal representations, admiralty computing service. Department of Scientific Research and Experiment, Admiralty

  13. Kythe PK (1998) Computational conformal mapping. Birkhäuser, Boston

    Google Scholar 

  14. Levin D, Papamichael N, Sideridis A (1977) On the use of conformal transformations for the numerical solution of harmonic boundary value problems. Comput Methods Appl Mech Eng 12: 201–218

    Article  MATH  MathSciNet  Google Scholar 

  15. Mathon R, Johnston RL (1977) The approximate solution of elliptic boundary-value problems by fundamental solutions. SIAM J Numer Anal 14: 638–650

    Article  MATH  MathSciNet  Google Scholar 

  16. Nehari Z (1952) Conformal mapping. Dover, New York

    MATH  Google Scholar 

  17. Papamichael N (2003) Dieter Gaier’s contributions to numerical conformal mapping. Comput Methods Funct Theory 3: 1–53

    MathSciNet  Google Scholar 

  18. Papamichael N, Stylianopoulos NS (2006) Numerical conformal mapping onto a rectangle. Department of Mathematics and Statistics, University of Cyprus (Preprint)

  19. Smyrlis Y-S (2008) Applicability and applications of the method of fundamental solutions. Math Comp (to appear)

  20. Smyrlis Y-S (2006) The method of fundamental solutions: a weighted least-squares approach. BIT 46: 164–193

    Article  MathSciNet  Google Scholar 

  21. Smyrlis Y-S, Karageorghis A (2001) Some aspects of the method of fundamental solutions for certain harmonic problems. J Sci Comput 16: 341–371

    Article  MathSciNet  Google Scholar 

  22. Smyrlis Y-S, Karageorghis A (2004) Numerical analysis of the MFS for certain harmonic problems. M2AN Math Model Numer Anal 38: 495–517

    Article  MathSciNet  Google Scholar 

  23. Spiegel MR (1964) Complex variables. McGraw-Hill, New York

    MATH  Google Scholar 

  24. Symm GT (1969) Conformal mapping of doubly-connected domains. Numer Math 13: 448–457

    Article  MATH  MathSciNet  Google Scholar 

  25. Tankelevich R, Fairweather G, Karageorghis A, Smyrlis Y-S (2006) Potential field based geometric modeling using the method of fundamental solutions. Int J Numer Methods Eng 68: 1257–1280

    Article  MathSciNet  Google Scholar 

  26. Trefethen LN (1989) SCPACK User’s Guide, Numerical Analysis Report 89-2, Department of Mathematics. MIT, Cambridge

  27. Tsangaris Th, Smyrlis Y-S, Karageorghis A (2005) Numerical analysis of the method of fundamental solutions for harmonic problems in annular domains. Numer Methods Partial Differ Equ 21: 507–539

    MathSciNet  Google Scholar 

  28. Warby MK (1992) BKMPACK User’s Guide, Technical Report, Department of Mathematics and Statistics. Brunel University, Uxbridge

  29. Wen G-C (1992) Conformal mappings and boundary value problems, Transactions of Mathematical Monographs, vol 106. American Mathematical Society, Providence

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Cyprus, 1678, Nicosia, Cyprus

    Andreas Karageorghis & Yiorgos-Sokratis Smyrlis

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  1. Andreas Karageorghis
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  2. Yiorgos-Sokratis Smyrlis
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Correspondence to Andreas Karageorghis.

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Karageorghis, A., Smyrlis, YS. Conformal mapping for the efficient MFS solution of Dirichlet boundary value problems. Computing 83, 1–24 (2008). https://doi.org/10.1007/s00607-008-0012-9

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