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

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Parallel Processing and Applied Mathematics (PPAM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12044))

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Abstract

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.

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Notes

  1. 1.

    There is also known directed interval arithmetic in which the left-ends of intervals may be greater than the right-end of ones. But it is not the case of our paper – we use only proper intervals.

References

  1. Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic Press, New York (1983)

    MATH  Google Scholar 

  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). https://doi.org/10.1007/978-3-642-78423-1

    Book  MATH  Google Scholar 

  3. Hansen, E.R.: Topics in Interval Analysis. Oxford University Press, London (1969)

    MATH  Google Scholar 

  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 5

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  8. Marciniak, A.: An interval difference method for solving the Poisson equation - the first approach. Pro Dialog 24, 49–61 (2008)

    Google Scholar 

  9. Marciniak, A.: Interval Arithmetic Unit (2016). http://www.cs.put.poznan.pl/amarciniak/IAUnits/IntervalArithmetic32and64.pas

  10. Marciniak, A.: Delphi Pascal Programs for Elliptic Boundary Value Problem (2019). http://www.cs.put.poznan.pl/amarciniak/IDM-EllipticEqn-Example

  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)

    Article  Google Scholar 

  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). https://doi.org/10.1007/978-3-319-75647-9_21

    Chapter  MATH  Google Scholar 

  13. Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)

    MATH  Google Scholar 

  14. Moore, R.E.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979)

    Book  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  16. Shokin, Y.I.: Interval Analysis. Nauka, Novosibirsk (1981)

    MATH  Google Scholar 

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Acknowledgments

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

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Authors and Affiliations

  1. Institute of Computing Science, Poznan University of Technology, Piotrowo 2, 60-965, Poznan, Poland

    Andrzej Marciniak

  2. Department of Computer Science, State University of Applied Sciences in Kalisz, Poznanska 201-205, 62-800, Kalisz, Poland

    Andrzej Marciniak

  3. Institute of Applied Mechanics, Poznan University of Technology, Jana Pawla II 24, 60-965, Poznan, Poland

    Malgorzata A. Jankowska

  4. Poznan Supercomputing and Networking Center, Jana Pawla II 10, 61-139, Poznan, Poland

    Tomasz Hoffmann

Authors
  1. Andrzej Marciniak
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  2. Malgorzata A. Jankowska
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  3. Tomasz Hoffmann
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Corresponding author

Correspondence to Andrzej Marciniak .

Editor information

Editors and Affiliations

  1. Czestochowa University of Technology, Czestochowa, Poland

    Roman Wyrzykowski

  2. University of Southern California, Marina del Rey, CA, USA

    Ewa Deelman

  3. University of Tennessee, Knoxville, TN, USA

    Jack Dongarra

  4. Czestochowa University of Technology, Czestochowa, Poland

    Konrad Karczewski

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Marciniak, A., Jankowska, M.A., Hoffmann, T. (2020). An Interval Difference Method of Second Order for Solving an Elliptical BVP. In: Wyrzykowski, R., Deelman, E., Dongarra, J., Karczewski, K. (eds) Parallel Processing and Applied Mathematics. PPAM 2019. Lecture Notes in Computer Science(), vol 12044. Springer, Cham. https://doi.org/10.1007/978-3-030-43222-5_36

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  • DOI: https://doi.org/10.1007/978-3-030-43222-5_36

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-43221-8

  • Online ISBN: 978-3-030-43222-5

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