The Maximum Norm Analysis of Schwarz Method for Elliptic Quasi-Variational Inequalities


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Authors: M. BEGGAS AND M. HAIOUR

DOI: 10.46793/KgJMat2104.635B

Abstract:

In this paper, we present a maximum norm analysis of an overlapping Schwartz method on non matching grids for a quasi-variational inequality, where the obstacle and the second member depend on the solution. Our result improves and generalizes some previous results.



Keywords:

Schwarz method, quasi-variational inequalities, weakly subsequenti ally continuous, L-error estimates.



References:

[1]   M. Boulbrachene, Optimal L-error estimate for variational inequalities with nonlinear source termms, Appl. Math. Lett. 15 (2002), 1013–1017.

[2]   M. Boulbrachene and S. Saadi, Maximum norm analysis of an over lapping non matching grids method for the obstacl problem, Adv. Difference Equ. 2006 (2006), Paper ID 085807.

[3]   M. Haiour and S. Boulaaras, Overlapping domain decomposition method for elliptic quasi-varational inqualities related to impulse control problem with mixed boundary conditions, Pro. Math. Sci. 121(4) (2011), 481–493.

[4]   M. Haiour and E. Hadidi, Uniform convergence of Schwarz method for varational inqualities for noncoercive variationam inequalities, International Journal of Contemporary Mathematical Sciences 4(28) (2009), 1423–1434.

[5]   J. Hannouzet and P. Joly, Convergence uniform des itérés définissant la solution d’une inéquation quasi-variantionnelle, C. R. Acad. Sci. Paris, Serie A 286 (1978).

[6]   P. L. Lions and P. Perthame, Une remarque sur les opérateurs nonlinéaire intervenant dand les inéquations quasi-variational, Ann. Fac. Sci. Toulouse Math. 5 (1983), 259–263.

[7]   P. L. Lions, On the Schwarz alternating method I, in: R. Gowinski, G. H. Golub, G. A. Meurant and J. Péeriaux (Eds.) Proceedings of First International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1988, 1–42.

[8]   P. L. Lions, On the Schwarz alternating method II, stochastic interpretation and order proprieties, domain decomposition methods, in: Proceedings of Second International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1989, 47–70.

[9]   P. A. Raviart, J. M. Thomas, Introduction á lAnalyse Numérique des Équations aux Dérivées Partielles, 3eme tirage, Masson, Paris, New York, Barcelone, 1992.

[10]   H. Mechri and S. Saadi, Overlapping nonmateching grid method for the ergodic control quasi-varational inequalities, American Journal of Computational Mathematics 3 (2013), 27–31.

[11]   S. Saadi and A. Mehri, L-error estimate of Schwarz algorithm for noncoercive variational inequalities, Appl. Math. Appl. 5(3) (2014), 572–580.

[12]   J. Zeng and S. Zhou, Schwarz altgorithm of the solution of variational inequalities with nonlinear source terms, Appl. Math. Comput. (1988), 23–35.