Nakao’s method and an interval difference scheme of second order for solving the elliptic BVP
Marciniak A. 1,2
1 Institute of Computing Science
Poznań University of Technology
Piotrowo 2, 60-965 Poznań, Poland
E-mail: andrzej.marciniak@put.poznan.pl2 Department of Computer Science
State University of Applied Sciences in Kalisz
Pozna´nska 201-205, 62-800 Kalisz, Poland
Received:
Received: 14 May 2019; revised: 11 June 2019; accepted: 26 June 2019; published online: 30 June 2019
DOI: 10.12921/cmst.2019.0000016
Abstract:
In the paper we compare Nakao’s method to our interval difference scheme of second order. Repeating some computational examples of Nakao, we have observed that our implementation of his method gives better results. Moreover, it appears that the presented interval difference scheme gives better enclosures of exact solutions than Nakao’s method. We also point out that the considered interval method can be used to solve the Poisson equation with Dirichlet’s condition, for which Nakao’s method is not applicable.
Key words:
elliptic boundary value problem, floating-point interval arithmetic, interval difference methods, Nakao’s method
References:
[1] R.E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs (1966).
[2] R.E. Moore, Methods and Applications of Interval Analysis, SIAM, Philadelphia (1979).
[3] G. Alefeld, J. Herzberger, Introduction to Interval Computations, Academic Press, New York (1983).
[4] R. Hammer, M. Hocks, U. Kulisch, D. Ratz, Numerical Toolbox for Verified Computing I. Basic Numerical Problems, Theory, Algorithms, and Pascal-XSC Programs, Springer- Verlag, Berlin (1993).
[5] E.R. Hansen, Topics in Interval Analysis, Oxford University Press, London (1969).
[6] M.T. Nakao, A Numerical Approach to the Proof of Existence of Solutions for Elliptic Problems, Japan Journal of Applied Mathematics 5, 313–332 (1988).
[7] M.T. Nakao, Solving Nonlinear Elliptic Problems with Result Verification Using H-1 Type Residual Iteration, Computing (Suppl.) 9, 161–173 (1993).
[8] M.T. Nakao, On Verified Computations of Solutions for Nonlinear Parabolic Problems, Nonlinear Theory and Its Applications. IEICE 5(3), 320–338 (2014).
[9] T. Meis, U. Marcowitz, Numerical Solution of Partial Differential Equations, Springer-Verlag, New York (1981).
[10] E. Süli, Lecture Notes on Finite Element Methods for Partial Differential Equations, University of Oxford (2000). https://people.maths.ox.ac.uk/suli/fem.pdf
[11] T. Hoffmann, A. Marciniak, Solving the Generalized Poisson Equation in Proper and Directed Interval Arithmetic, Computational Methods in Science and Technology 22(4), 225– 232 (2016).
[12] T. Hoffmann, A. Marciniak, B. Szyszka, Interval Versions of Central Difference Method for Solving the Poisson Equation in Proper and Directed Interval Arithmetic, Foundations of Computing and Decision Sciences 38(3), 193–206 (2013).
[13] T. Hoffmann, A. Marciniak, Solving the Poisson Equation by an Interval Method of the Second Order, Computational Methods in Science and Technology 19(1), 13–21 (2013).
[14] A. Marciniak, T. Hoffmann, Interval Difference Methods for Solving the Poisson Equation, [In:] S. Pinelas, T. Caraballo, P. Kloeden, J.R. Graef (eds.), Differential and Difference Equations with Applications, vol. 230 of Springer Proceedings in Mathematics & Statistics, 259–270, Springer (2018).
[15] A. Marciniak, An Interval Difference Method for Solving the Poisson Equation – the First Approach, Pro Dialog 24, 49–61 (2008).
[16] A. Marciniak, Interval Arithmetic Unit, (2016). http://www.cs.put.poznan.pl/amarciniak/IAUnits/IntervalArithmetic32and64.pas
[17] R.E. Moore, R.B. Kearfott, M.J. Cloud, Introduction to Interval Analysis, SIAM, Philadelphia (2009).
[18] A. Marciniak, Delphi Pascal Programs for Nakao and Interval Difference Methods for Solving the Elliptic BVP, (2019). http://www.cs.put.poznan.pl/amarciniak/NIDM-EllipticBVP-Examples
In the paper we compare Nakao’s method to our interval difference scheme of second order. Repeating some computational examples of Nakao, we have observed that our implementation of his method gives better results. Moreover, it appears that the presented interval difference scheme gives better enclosures of exact solutions than Nakao’s method. We also point out that the considered interval method can be used to solve the Poisson equation with Dirichlet’s condition, for which Nakao’s method is not applicable.
Key words:
elliptic boundary value problem, floating-point interval arithmetic, interval difference methods, Nakao’s method
References:
[1] R.E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs (1966).
[2] R.E. Moore, Methods and Applications of Interval Analysis, SIAM, Philadelphia (1979).
[3] G. Alefeld, J. Herzberger, Introduction to Interval Computations, Academic Press, New York (1983).
[4] R. Hammer, M. Hocks, U. Kulisch, D. Ratz, Numerical Toolbox for Verified Computing I. Basic Numerical Problems, Theory, Algorithms, and Pascal-XSC Programs, Springer- Verlag, Berlin (1993).
[5] E.R. Hansen, Topics in Interval Analysis, Oxford University Press, London (1969).
[6] M.T. Nakao, A Numerical Approach to the Proof of Existence of Solutions for Elliptic Problems, Japan Journal of Applied Mathematics 5, 313–332 (1988).
[7] M.T. Nakao, Solving Nonlinear Elliptic Problems with Result Verification Using H-1 Type Residual Iteration, Computing (Suppl.) 9, 161–173 (1993).
[8] M.T. Nakao, On Verified Computations of Solutions for Nonlinear Parabolic Problems, Nonlinear Theory and Its Applications. IEICE 5(3), 320–338 (2014).
[9] T. Meis, U. Marcowitz, Numerical Solution of Partial Differential Equations, Springer-Verlag, New York (1981).
[10] E. Süli, Lecture Notes on Finite Element Methods for Partial Differential Equations, University of Oxford (2000). https://people.maths.ox.ac.uk/suli/fem.pdf
[11] T. Hoffmann, A. Marciniak, Solving the Generalized Poisson Equation in Proper and Directed Interval Arithmetic, Computational Methods in Science and Technology 22(4), 225– 232 (2016).
[12] T. Hoffmann, A. Marciniak, B. Szyszka, Interval Versions of Central Difference Method for Solving the Poisson Equation in Proper and Directed Interval Arithmetic, Foundations of Computing and Decision Sciences 38(3), 193–206 (2013).
[13] T. Hoffmann, A. Marciniak, Solving the Poisson Equation by an Interval Method of the Second Order, Computational Methods in Science and Technology 19(1), 13–21 (2013).
[14] A. Marciniak, T. Hoffmann, Interval Difference Methods for Solving the Poisson Equation, [In:] S. Pinelas, T. Caraballo, P. Kloeden, J.R. Graef (eds.), Differential and Difference Equations with Applications, vol. 230 of Springer Proceedings in Mathematics & Statistics, 259–270, Springer (2018).
[15] A. Marciniak, An Interval Difference Method for Solving the Poisson Equation – the First Approach, Pro Dialog 24, 49–61 (2008).
[16] A. Marciniak, Interval Arithmetic Unit, (2016). http://www.cs.put.poznan.pl/amarciniak/IAUnits/IntervalArithmetic32and64.pas
[17] R.E. Moore, R.B. Kearfott, M.J. Cloud, Introduction to Interval Analysis, SIAM, Philadelphia (2009).
[18] A. Marciniak, Delphi Pascal Programs for Nakao and Interval Difference Methods for Solving the Elliptic BVP, (2019). http://www.cs.put.poznan.pl/amarciniak/NIDM-EllipticBVP-Examples
