Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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VERIFIED COMPUTATIONS OF SOLUTIONS FOR SOME UNILATERAL BOUNDARY VALUE PROBLEMS FOR SECOND ORDER EQUATIONS

  • Received : 2021.03.31
  • Accepted : 2021.05.03
  • Published : 2021.05.30
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Abstract

In this paper, we propose a new iterative algorithm to automatically prove the existence of solutions for a unilateral boundary value problems for second order equations.

Keywords

Acknowledgement

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No. 2017R1A2B4006092).

References

  1. N.W. Bazley, D.W. Fox, Comparision operators for lower bounds to eigenvalues, J. reine angew. Math. 223 (1966), 142-149.
  2. J. Cea, Optimisation, theorie et algorithmes, Paris: Dunod, 1971.
  3. X. Chen, A verification method for solutions of nonsmooth equations, Computing 58 (1997), 281-294. https://doi.org/10.1007/BF02684394
  4. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
  5. R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer, New York, 1984.
  6. I. Hlavacek, J. Haslinger, J. Necas, J. Lovisek, Solution of Variational Inequalities in Mechanics, Springer Ser. Appl. Math. Sci. 66 (1988), 221-265.
  7. T. Kato, On the upper and lower bounds of eigenvalues, J. Phys. Soc. Japan 4 (1949), 334-339. https://doi.org/10.1143/JPSJ.4.334
  8. N.J. Lehmann, Optimale Eigenwerteinschlieungen, Numer. Math. 5 (1963), 246-272. https://doi.org/10.1007/BF01385896
  9. U. Mosco, Approximation of the solutions of some variational inequalities, Roma: Press of Roma University, 1971.
  10. M.T. Nakao, A numerical verification method for the existence of weak solutions for nonlinear boundary value problems, Journal of Math. Analysis and Appl. 164 (1992), 489-507. https://doi.org/10.1016/0022-247x(92)90129-2
  11. Y. Watanabe, N. Yamamoto, M.T. Nakao, Verified computations of solutions for nondifferential elliptic equations related to MHD equilibria. Nonlinear Analysis, Theory, Mathods & Applications 28 (1997), 577-587. https://doi.org/10.1016/0362-546X(95)00165-R
  12. F. Natterer, Optimale L2-Konvergenz finiter Elemente bei Variationsungleichungen, Bonn. Math. Schr. 89 (1976), 1-12.
  13. M. Plum, Explict H2-Estimates and Pointwise Bounds for Solutions of Second-Order Elliptic Bounday Value Problems. Journal of Mathematical Analysis and Appl. 165 (1992), 36-61. https://doi.org/10.1016/0022-247x(92)90067-n
  14. M. Plum, Enclosures for weak solutions of nonlinear elliptic boundary value problems, WSSIAA, World Scientific Publishing Company 3 (1994), 505-521.
  15. M. Plum, Existence and Enclosure Results for Continua of Solutions of Parameter - Dependent Nonlinear Boundary Value Problems, Journal of Computational and Applied Mathematics 60 (1995), 187-200. https://doi.org/10.1016/0377-0427(94)00091-E
  16. S.M. Rump, Solving algebraic problems with high accuracy, A new approach to scientific computation (Edited by U.W. KULISH and W.L. MIRANKER), Academic Press, New York, 1983.
  17. C.S. Ryoo, Numerical verification of solutions for a simplified Signorini problem, Comput. Math. Appl. 40 (2000), 1003-1013. https://doi.org/10.1016/S0898-1221(00)85011-7
  18. C.S. Ryoo, An approach to the numerical verification of solutions for obstacle problems, Comput. Math. Appl. 53 (2007), 842-850. https://doi.org/10.1016/j.camwa.2006.03.042
  19. C.S. Ryoo, Numerical verification of solutions for Signorini problems using Newton-like mathod, International Journal for Numerical Methods in Engineering 73 (2008), 1181-1196. https://doi.org/10.1002/nme.2121
  20. C.S. Ryoo and R.P. Agarwal, Numerical inclusion methods of solutions for variational inequalities, International Journal for Numerical Methods in Engineering 54 (2002), 1535-1556. https://doi.org/10.1002/nme.479
  21. C.S. Ryoo and R.P. Agarwal, Numerical verification of solutions for generalized obstacle problems, Neural, Parallel & Scientific Compuataions 11 (2003), 297-314.
  22. C.S. Ryoo and M.T. Nakao, Numerical verification of solutions for variational inequalities, Numerische Mathematik 81 (1998), 305-320. https://doi.org/10.1007/s002110050394
  23. F. Scarpini, M.A. Vivaldi, Error estimates for the approximation of some unilateral problems, R.A.I.R.O. Numerical Analysis 11 (1977), 197-208.
  24. G. Strang, G. Fix, An analysis of the finite element method, Englewood Cliffs, New Jersey, Prentice-Hall, 1973.