Asymptotic Expansions for the Discretization Error In Poisson’s Equation on General Domains

  • Klaus Böhmer
Part of the ISNM International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’Analyse Numérique book series (ISNM, volume 51)


In this paper we modify the usual five-point-difference discretization near the boundary of a general domain as to guarantee the existence of an asymptotic expansion. We generalize and improve results due to Gerschgorin [9], Collatz [8], Mikeladse [11], Wasow [14] and Pereyra-Proskurowski-Widlund [12].


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  1. [1]
    Agmon, S., A. Douglis, L. Nirenberg: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math. 12, 623–727 (1959).CrossRefGoogle Scholar
  2. [2]
    Bickley, W.G.: Formulae for numerical differentiation, Mathematical Gazette 25, 19–27 (1959).CrossRefGoogle Scholar
  3. [3]
    Böhmer, K.: Discrete Newton methods and iterated defect corrections, I General theory. II Proofs and applications to initial and boundary value problems, submitted to Numer. Math.Google Scholar
  4. [4]
    Böhmer, K.: High order difference methods for quasilinear elliptic boundary value problems on general regions, Univers. of Wisconsin-Madison, MRC, Technical Summary Report 1979.Google Scholar
  5. [5]
    Böhmer, K.: Asymptotic expansion for the discretization error in linear elliptic boundary value problems on general domains.Google Scholar
  6. [6]
    Brakhage, H.: Über die numerische Behandlung von Integralgleichungen nach der Quadraturformelmethode, Numer. Math. 2, 183–196 (1960)CrossRefGoogle Scholar
  7. [7]
    Bramble, J.H., B.E. Hubbard: A theorem on error estimation for finite difference analogues of the Dirichlet problem for elliptic equations, Contrib. Diff. Equat. 2, 319–340 (1963).Google Scholar
  8. [8]
    Collatz, L.: Bemerkungen zur Fehlerabschätzung für das Differenzenverfahren bei partiellen Differentialgleichungen, Z. Angew. Math. Mech. 13, 56–57 (1933).CrossRefGoogle Scholar
  9. [9]
    Gerschgorin, S.: Eehlerabschätzung für das Differenzenverfahren zur Lösung partieller Differentialgleichungen, Z. Angew. Math. Mech. 10, 373–382 (1920).CrossRefGoogle Scholar
  10. [10]
    Hofmann, P.: Asymptotic expansions of the discretization error of boundary value problems of the Laplace equation in rectangular domains, Numer. Math. 9, 302–322 (1967).CrossRefGoogle Scholar
  11. [11]
    Mikeladse, S.E.: Neue Methoden der Integration von elliptischen und parabolischen Differentialgleichungen, Izv. Akad. Nauk SSSR, Seria Mat. 5, 57–74 (1941) (russian).Google Scholar
  12. [12]
    Pereyra, V., W. Proskurowski and O. Widlund: High order fast Laplace solvers for the Dirichlet problem on general regions, Math. Comp. 31, 1–16 (1977).CrossRefGoogle Scholar
  13. [13]
    Shortley, G., R. Weiler: The numerical solution of Laplace’s equation, J. Appl. Phys. 9, 334–348 (1938).CrossRefGoogle Scholar
  14. [14]
    Wasow, W.: Discrete approximations to elliptic differential equations, Z. Angew. Math. Phys. 6, 81–97 (1955).CrossRefGoogle Scholar
  15. [15]
    Zenger, C., H. Gietl: Improved difference schemes for the Dirichlet problem of Poisson’s equation in the neighbourhood of corners, Numer. Math. 30, 315–332 (1978).CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1979

Authors and Affiliations

  • Klaus Böhmer
    • 1
  1. 1.Institut für Praktische MathematikUniversität KarlsruheKarlsruheW.-Germany

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