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**A Cubic Spline Method for Solving a Unilateral Obstacle Problem** ()

Department of Mathematics and Computer Science, Faculty of Science, University Mohammed Premier, Oujda, Morocco.

MATSI Laboratory, ESTO, University Mohammed Premier, Oujda, Morocco.

Department of Mathematics, Faculty of Science and Technology,University Sultan Moulay Slimane, Beni-Mellal, Morocco.

Department of Mathematics, Faculty of Science and Technology, University Sultan Moulay Slimane, Beni-Mellal, Morocco.

MATSI Laboratory, ESTO, University Mohammed Premier, Oujda, Morocco.

Department of Mathematics, Faculty of Science and Technology,University Sultan Moulay Slimane, Beni-Mellal, Morocco.

Department of Mathematics, Faculty of Science and Technology, University Sultan Moulay Slimane, Beni-Mellal, Morocco.

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.Share and Cite:

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.Conflicts of Interest

The authors declare no conflicts of interest.

[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 |

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