An Interval Difference Method of Second Order for Solving an Elliptical BVP

  • Andrzej MarciniakEmail author
  • Malgorzata A. Jankowska
  • Tomasz Hoffmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12044)


In the article we present an interval difference scheme for solving a general elliptic boundary value problem with Dirichlet’ boundary conditions. The obtained interval enclosure of the solution contains all possible numerical errors. A numerical example we present confirms that the exact solution belongs to the resulting interval enclosure.


Interval difference methods Elliptic boundary value problem Floating-point interval arithmetic 



The paper was supported by the Poznan University of Technology (Poland) through the Grants No. 09/91/DSPB/1649 and 02/21/ SBAD/3558.


  1. 1.
    Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic Press, New York (1983)zbMATHGoogle Scholar
  2. 2.
    Hammer, R., Hocks, M., Kulisch, U., Ratz, D.: Numerical Toolbox for Verified Computing I. Basic Numerical Problems, Theory, Algorithms, and Pascal-XSC Programs. Springer, Berlin (1993). Scholar
  3. 3.
    Hansen, E.R.: Topics in Interval Analysis. Oxford University Press, London (1969)zbMATHGoogle Scholar
  4. 4.
    Hoffmann, T., Marciniak, A.: Finding optimal numerical solutions in interval versions of central-difference method for solving the poisson equation. In: Łatuszyńska, M., Nermend, K. (eds.) Data Analysis - Selected Problems, pp. 79–88. Scientific Papers of the Polish Information Processing Society Scientific Council, Szczecin-Warsaw (2013). Chapter 5Google Scholar
  5. 5.
    Hoffmann, T., Marciniak, A.: Solving the Poisson equation by an interval method of the second order. Comput. Methods Sci. Technol. 19(1), 13–21 (2013)CrossRefGoogle Scholar
  6. 6.
    Hoffmann, T., Marciniak, A.: Solving the generalized poisson equation in proper and directed interval arithmetic. Comput. Methods Sci. Technol. 22(4), 225–232 (2016)CrossRefGoogle Scholar
  7. 7.
    Hoffmann, T., Marciniak, A., Szyszka, B.: Interval versions of central difference method for solving the Poisson equation in proper and directed interval arithmetic. Found. Comput. Decis. Sci. 38(3), 193–206 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Marciniak, A.: An interval difference method for solving the Poisson equation - the first approach. Pro Dialog 24, 49–61 (2008)Google Scholar
  9. 9.
  10. 10.
    Marciniak, A.: Delphi Pascal Programs for Elliptic Boundary Value Problem (2019).
  11. 11.
    Marciniak, A.: Nakao’s method and an interval difference scheme of second order for solving the elliptic BVS. Comput. Methods Sci. Technol. 25(2), 81–97 (2019)CrossRefGoogle Scholar
  12. 12.
    Marciniak, A., Hoffmann, T.: Interval difference methods for solving the Poisson equation. In: Pinelas, S., Caraballo, T., Kloeden, P., Graef, J.R. (eds.) ICDDEA 2017. SPMS, vol. 230, pp. 259–270. Springer, Cham (2018). Scholar
  13. 13.
    Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)zbMATHGoogle Scholar
  14. 14.
    Moore, R.E.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979)CrossRefGoogle Scholar
  15. 15.
    Nakao, M.T.: A numerical approach to the proof of existence of solutions for elliptic problems. Japan J. Appl. Math. 5, 313–332 (1988)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Shokin, Y.I.: Interval Analysis. Nauka, Novosibirsk (1981)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Andrzej Marciniak
    • 1
    • 2
    Email author
  • Malgorzata A. Jankowska
    • 3
  • Tomasz Hoffmann
    • 4
  1. 1.Institute of Computing SciencePoznan University of TechnologyPoznanPoland
  2. 2.Department of Computer ScienceState University of Applied Sciences in KaliszKaliszPoland
  3. 3.Institute of Applied MechanicsPoznan University of TechnologyPoznanPoland
  4. 4.Poznan Supercomputing and Networking CenterPoznanPoland

Personalised recommendations