Advertisement

Numerische Mathematik

, Volume 141, Issue 3, pp 605–626 | Cite as

Solution of the Dirichlet problem by a finite difference analog of the boundary integral equation

  • J. Thomas BealeEmail author
  • Wenjun Ying
Article
  • 46 Downloads

Abstract

Several important problems in partial differential equations can be formulated as integral equations. Often the integral operator defines the solution of an elliptic problem with specified jump conditions at an interface. In principle the integral equation can be solved by replacing the integral operator with a finite difference calculation on a regular grid. A practical method of this type has been developed by the second author. In this paper we prove the validity of a simplified version of this method for the Dirichlet problem in a general domain in \({\mathbb {R}}^2\) or \({\mathbb {R}}^3\). Given a boundary value, we solve for a discrete version of the density of the double layer potential using a low order interface method. It produces the Shortley–Weller solution for the unknown harmonic function with accuracy \(O(h^2)\). We prove the unique solvability for the density, with bounds in norms based on the energy or Dirichlet norm, using techniques which mimic those of exact potentials. The analysis reveals that this crude method maintains much of the mathematical structure of the classical integral equation. Examples are included.

Mathematics Subject Classification

31C20 35J05 45B05 65N06 65N12 

Notes

References

  1. 1.
    Beale, J.T., Layton, A.T.: On the accuracy of finite difference methods for elliptic problems with interfaces. Commun. Appl. Math. Comput. Sci. 1, 91–119 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ciarlet, P.G.: Discrete maximum principle for finite-difference operators. Aequationes Math. 4, 338–352 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Costabel, M.: Some historical remarks on the positivity of boundary integral operators. In: Schanz, M., Steinbach, O. (eds.) Boundary element analysis. Lecture notes in applied and computational mechanics, vol. 29, pp. 1–27. Springer, Berlin (2007)CrossRefGoogle Scholar
  4. 4.
    Forsythe, G.E., Wasow, W.R.: Finite-Difference Methods for Partial Differential Equations. Wiley, New York (1960)zbMATHGoogle Scholar
  5. 5.
    Hackbusch, W.: Elliptic Differential Equations: Theory and Numerical Treatment, 2nd edn. Springer, Berlin (2017)CrossRefzbMATHGoogle Scholar
  6. 6.
    Hommel, A.: Fundamentallösungen partieller Dierenzenoperatoren und die Lösung diskreter Randwertprobleme mit Hilfe von Dierenzenpotentialen. Dissertation, Bauhaus-Universität Weimar (1998)Google Scholar
  7. 7.
    Hommel, A.: Solution of Dirichlet problems with discrete double-layer potentials. In: Proceedings of the Eighth International Conference on Difference Equations and Applications, pp. 171–178. Chapman & Hall, Boca Raton (2005)Google Scholar
  8. 8.
    Ito, K., Lai, M.-C., Li, Z.: A well-conditioned augmented system for solving Navier–Stokes equations in irregular domains. J. Comput. Phys. 228, 2616–2628 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jomaa, Z., Macaskill, C.: The embedded finite difference method for the Poisson equation in a domain with an irregular boundary and Dirichlet boundary conditions. J. Comput. Phys. 202, 488–506 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Li, Z., Ito, K.: The Immersed Interface Method. SIAM, Philadelphia (2006)CrossRefzbMATHGoogle Scholar
  11. 11.
    Liu, X.-D., Fedkiw, R.P., Kang, M.: A boundary condition capturing method for Poisson’s equation on irregular domains. J. Comput. Phys. 160, 151–178 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liu, X.-D., Sideris, T.C.: Convergence of the ghost fluid method for elliptic equations with interfaces. Math. Comput. 72, 1731–46 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Matsunaga, M., Yamamoto, T.: Superconvergence of the ShortleyWeller approximation for Dirichlet problems. J. Comput. Appl. Math. 116, 263–73 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mayo, A.: The fast solution of Poisson’s and the biharmonic equations on irregular regions. SIAM J. Numer. Anal. 21, 285–299 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Medvinsky, M., Tsynkov, S., Turkel, E.: The method of difference potentials for the Helmholtz equation using compact high order schemes. J. Sci. Comput. 53, 150–93 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Proskurowski, W., Widlund, O.: On the numerical solution of Helmholtz’s equation by the capacitance matrix method. Math. Comput. 30, 433–468 (1976)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ryabenkii, V.S.: The Method of Difference Potentials and Its Applications. Springer, Berlin (2002)CrossRefGoogle Scholar
  18. 18.
    Samarskii, A.A.: The Theory of Difference Schemes. M. Decker, New York (2001)CrossRefzbMATHGoogle Scholar
  19. 19.
    Steinbach, O., Wendland, W.L.: On C. Neumann’s method for second-order elliptic systems in domains with non-smooth boundaries. J. Math. Anal. Appl. 262, 733–48 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Weynans, L.: Super-convergence in maximum norm of the gradient for the Shortley–Weller method. J. Sci. Comput. (2017).  https://doi.org/10.1007/s10915-017-0548-y
  21. 21.
    Wiegmann, A., Bube, K.P.: The explicit-jump immersed interface method: finite difference methods for PDEs with piecewise smooth solutions. SIAM J. Numer. Anal. 37, 827–62 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ying, W.-J., Henriquez, C.S.: A kernel-free boundary integral method for elliptic boundary value problems. J. Comput. Phys. 227, 1046–1074 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ying, W.-J., Wang, W.-C.: A kernel-free boundary integral method for implicitly defined surfaces. J. Comput. Phys. 252, 606–624 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ying, W.-J., Wang, W.-C.: A kernel-free boundary integral method for variable coefficients elliptic PDEs. Commun. Comput. Phys. 15, 1108–1140 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ying, W.-J.: A kernel-free boundary integral method for the nonlinear Poisson–Boltzmann equation (submitted)Google Scholar
  26. 26.
    Yoon, G., Min, C.: Convergence analysis of the standard central finite difference method for Poisson equation. J. Sci. Comput. 67, 602–17 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA
  2. 2.School of Mathematical Sciences, MOE-LSC and Institute of Natural SciencesShanghai Jiao Tong UniversityMinhang, ShanghaiPeople’s Republic of China

Personalised recommendations