AJCM> Vol.2 No.3, September 2012

A Cubic Spline Method for Solving a Unilateral Obstacle Problem

DownloadDownload as PDF (Size:228KB) Full-Text HTML PP. 217-222   DOI: 10.4236/ajcm.2012.23028

ABSTRACT

This paper, we develop a numerical method for solving a unilateral obstacle problem by using the cubic spline collocation method and the generalized Newton method. This method converges quadratically if a relation-ship between the penalty parameter and the discretization parameter h is satisfied. An error estimate between the penalty solution and the discret penalty solution is provided. To validate the theoretical results, some numerical tests on one dimensional obstacle problem are presented.

KEYWORDS


Cite this paper

E. Mermri, A. Serghini, A. El hajaji and K. Hilal, "A Cubic Spline Method for Solving a Unilateral Obstacle Problem," American Journal of Computational Mathematics, Vol. 2 No. 3, 2012, pp. 217-222. doi: 10.4236/ajcm.2012.23028.

References

[1] R. Glowinski, J.L. Lions, R. Trémolières, Numerical analysis of variational inequalities, North-Holland, Amsterdam, 1981.
[2] D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, New York, 1980.
[3] R.P. Agarwal, C.S. Ryoo, Numerical verifications of solutions for obstacle problems, Computing Suppl, Vol. 15, 2001, pp. 9-19.
[4] R. Glowinski, Y.A. Kuznetsov, T-W. Pan, A penalty/Newton/conjugate gradient method for the solution of obstacle problems, C. R. Acad. Sci. Paris, Ser, Vol. 1336, 2003, pp. 435-440.
[5] H. Huang, W. Han, J. Zhou, The regularisation method for an obstacle problem, Numer. Math, Vol. 69, 1994, pp. 155-166. doi:10.1007/s002110050086
[6] R. Scholz, Numerical solution of the Obstacle problem by the penalty method, Computing, Vol. 32, 1984, pp 297-306. doi:10.1007/BF02243774
[7] H. Lewy, G. Stampacchia, On the regularity of the solution of the variational inequalities, Communications in Pure and Applied Mathematics, Vol. 22, 1969, pp. 153-188. doi:10.1002/cpa.3160220203
[8] X. Chen, A verification method for solutions of nonsmooth equations, Computing, Vol. 58, 1997, pp. 281-294. doi:10.1007/BF02684394
[9] X. Chen, Z. Nashed, L. Qi, Smooting methods and semismooth methods for nondiffentiable operator equations, SIAM J. Numer. Anal, Vol. 38, No. 4, 2000, pp. 1200-1216. doi:10.1137/S0036142999356719
[10] M.J. ?mietański, A generalizd Jacobian based Newton method for semismooth block-triangular system of equations, Journal of Computational and Applied Mathematics, Vol. 205, 2007, pp. 305-313. doi:10.1016/j.cam.2006.05.003
[11] H. N. ?aglar, S. N. ?aglar, E. H. Twizell, The numerical solution of fifth-order boundary value problems with sixth-degree B-spline function, Applied Mathematics Letter, Vol. 12, 1999, pp. 25-30. doi:10.1016/S0893-9659(99)00052-X
[12] A. Lamnii, H. Mraoui, D. Sbibih, A. Tijini, Sextic Spline Solution of Nonlinear Fifth-Order Boundary Value Problems, Inter. J. Comput. Math, Vol. 88, 2011, pp. 2072-2088. doi:10.1080/00207160.2010.519384
[13] C. de Boor, A Practical guide to Splines. Springer Verlag, NewYork, 1978.
[14] R.R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, Springer, 1993.
[15] F.H. Clarke, Optimization and nonsmooth analysis, New York, Wiley, 1993.
[16] L. Qi, Convergence analysis of some algorithms for solving some nonsmooth nonsmooth equations, Math. Oper. Res, Vol. 18, 1993, pp. 227-244. doi:10.1287/moor.18.1.227
[17] L. Qi, J. Sun, A nonsmooth version of the Newthon's method, Math. Programming, Vol. 58, 1993, pp. 353-367. doi:10.1007/BF01581275
[18] J.S. Pang and L. Qi, Nonsmooth functions: Motivation and algoritms, SIAM J. Optimization, Vol. 3, 1993, pp. 443-465. doi:10.1137/0803021

comments powered by Disqus

Copyright © 2014 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.