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L-Error Estimate of Schwarz Algorithm for Noncoercive Variational Inequalities

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DOI: 10.4236/am.2014.53054    3,109 Downloads   4,425 Views   Citations


The Schwarz method for a class of elliptic variational inequalities with noncoercive operator was studied in this work. The author proved the error estimate in L∞-norm for two domains with overlapping nonmatching grids using the geometrical convergence of solutions and the uniform convergence of subsolutions.

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S. Saadi and A. Mehri, "L-Error Estimate of Schwarz Algorithm for Noncoercive Variational Inequalities," Applied Mathematics, Vol. 5 No. 3, 2014, pp. 572-580. doi: 10.4236/am.2014.53054.


[1] P. L. Lions, “On the Schwarz Alternating Method. I,” 1st International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1988, pp. 1-42.
[2] P. L. Lions, “On the Schwarz Alternating Method. II, Stochastic Interpretation and Order Proprieties,” In: T. F. Chan, et al., Eds., Domain Decomposition Methods, SIAM, Philadelphia, 1989, pp. 47-70.
[3] L. Badea, “On the Schwarz Alternating Method with More than Subdomains for Nonlinear Monotone Problems,” SIAM Journal of Numerical Analysis, Vol. 28, 1991, pp. 197-204.
[4] L. Badea, X. Cheng and J. Wang, “Convergence Rate Analysis of a Multiplicative Schwarz Method for Variational Inequalities,” SIAM Journal of Numerical Analysis, Vol. 41, No. 3, 2003, pp. 1052-1073.
[5] M. Boulbrachene and S. Saadi, “Maximum Norm Analysis of an Overlapping Nonmatching Grids Method for the Obstacle Problem,” Hindawi Publishing Corporation, Cairo, 2006, pp. 1-10.
[6] M. Haiour and E. Hadidi, “Uniform Convergence of Schwarz Method for Noncoercive Variational Inequalities,” International Journal of Contemporary Mathematical Sciences, Vol. 4, No. 29, 2009, pp. 1423-1434.
[7] M. Haiour and S. Boulaares, “Overlapping Domain Decomposition Methods for Elliptic Quasi-Variational Inequalities Related to Impulse Control Problem with Mixed Boundary Conditions,” Proceedings of Indian Academy of Sciences (Mathematical Sciences), Vol. 121, No. 4, 2011, pp. 481-493.
[8] J. Zeng and S. Zhou, “On Monotone and Geometric Convergence of Schwarz Methods for Two-Sided Obstacle Problems,” SIAM, Journal on Numerical Analysis, Vol. 35, No. 2, 1998, pp. 600-616.
[9] J. Zeng and S. Zhou, “Schwarz Algorithm of the Solution of Variational Inequalities with Nonlinear Source Terms,” Applied Mathematics and Computations, Vol. 97, 1998, pp. 23-35.
[10] A. Bensoussan and J. L. Lions, “Applications of Variational Inequalities in Stochastic Control (English Version),” North-Holland Publishing Company, Amsterdam, 1982.
[11] M. Boulbrachene, “The Noncoercive Quasi-Variational Inequalities Related to Impulse Control Problems,” Computers & Mathematics with Applications, Vol. 35, No. 12, 1998, pp. 101-108.
[12] P. Cortey-Dumont, “On Finite Element Approximation in the L∞-Norm of Variational Inequalities,” Numerische Mathematik, Vol. 47, No. 1, 1985, pp. 45-57.
[13] P. G. Ciarlet and P. A. Raviart, “Maximum Principle and Uniform Convergence for the Finite Element Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 2, 1973, pp. 1-20.
[14] P. Cortey-Dumont, “Sur les Inéquations Variationnelles à Opérateur non Coercif,” Rairo, Modélisation Mathématique Et Analyse Numérique, Vol. T.19, No. 2, 1985, pp. 195-212.

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