Affiliation(s)

Laboratory LANOS, Department of Mathematics, Badji Mokhtar University, Annaba, Algeria.

Laboratory LAIG, Department of Mathematics, Guelma University, Guelma, Algeria.

Laboratory LANOS, Department of Mathematics, Badji Mokhtar University, Annaba, Algeria.

Laboratory LAIG, Department of Mathematics, Guelma University, Guelma, Algeria.

ABSTRACT

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.

Cite this paper

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.

S. Saadi and A. Mehri, "

References

[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.

http://dx.doi.org/10.1137/S0036142901393607

[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. http://dx.doi.org/10.1137/S0036142995288920

[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. http://dx.doi.org/10.1016/S0898-1221(98)00100-X

[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. http://dx.doi.org/10.1007/BF01389875

[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.

http://www.numdam.org/numdam-bin/feuilleter?id=M2AN_1985__19_2

[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.

http://dx.doi.org/10.1137/S0036142901393607

[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. http://dx.doi.org/10.1137/S0036142995288920

[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. http://dx.doi.org/10.1016/S0898-1221(98)00100-X

[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. http://dx.doi.org/10.1007/BF01389875

[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.

http://www.numdam.org/numdam-bin/feuilleter?id=M2AN_1985__19_2