The maximum norm analysis of a nonmatching grids method for a class of parabolic $p(x)$-Laplacien equation

  • Sadok Otmani University of El Oued
  • Salah Boulaaras Qassim University
  • Ali Allahem Qassim University

Résumé

Motivated by the work of Boulaaras and Haiour in [7], we provide a maximum norm analysis of Schwarz alternating method for parabolic p(x)-Laplacien equation, where an optimal error analysis each subdomain between the discrete Schwarz sequence and the continuous solution of the presented problem is established

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Références

L. Badea. On the schwarz alternating method with more than two subd-omains for monotone problems. SIAM Journal on Numerical Analysis 28 (1991), no. 1, 179-204. https://doi.org/10.1137/0728010

A. Bensoussan & J. L. Lions. Contrˆole impulsionnel et In 'equations Quasi-variationnelles. Dunod, 1982.

S. Boulaaras, M. Haiour, "L∞ -asymptotic behavior for a finite element approximation in parabolic quasi-variational inequalities related to impulse control problem", App. Math. Comp., 217, 6443-6450 (2011). https://doi.org/10.1016/j.amc.2011.01.025

Cai, X C, Mathew, T P, Sarkis, M V: Maximum norm analysis of overlapping nonmatching grid discretization of elliptic equations. SIAM J. Numer. Anal. 5, 1709-1728 (2000) https://doi.org/10.1137/S0036142998348741

J. Douglas, Jr. and C-S. Huang, An accelerated domain decomposition procedure based on Robin transmissionconditions, BIT Numerical Mathematics, vol. 37, no. 3, pp. 678-686, 1997. https://doi.org/10.1007/BF02510246

B. Engquist and H. -K. Zhao, Absorbing boundary conditions for domain decomposition, Applied Numerical Mathematics, vol. 27, no. 4, pp. 341-365, 1998. https://doi.org/10.1016/S0168-9274(98)00019-1

M. Haiour, S. Boulaaras, Overlapping domain decomposition methods for elliptic quasi-variational inequalities related to impulse control problem with mixed boundary conditions, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 121,No. 4, November 2011,pp.481-493 https://doi.org/10.1007/s12044-011-0042-7

P.-L. Lions, On the Schwarz alternating method. I, in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. H. Golub, G. A. Meurant, and J.P ' eriaux,Eds.,pp. 1-42, SIAM, Philadelphia, USA,1988.

T. F. Chan, T. Y. Hou, and P.-L. Lions, Geometry related convergence results for domain decomposition algorithms, SIAM Journal on Numerical Analysis, vol. 28, no. 2, pp. 378-391, 1991. https://doi.org/10.1137/0728021

A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, The Clarend on Press,Oxford, UK, 1999. https://doi.org/10.1007/978-94-011-4647-0_11

A. Toselli and O. Widlund, Domain Decomposition Methods Algorithms and Theory, vol. 34 of Springer Seriesin Computational Mathematics, Springer, Berlin, Germany, 2005. https://doi.org/10.1007/b137868

Y. Maday and F. Magoul'es, Improved ad hoc interface conditions for Schwarz solution procedure tuned to highly heterogeneous media, Applied Mathematical Modelling, vol. 30, no. 8, pp. 731-743, 2006. https://doi.org/10.1016/j.apm.2005.05.020

Y. Maday and F. Magoul'es, A survey of various absorbing interface conditions for the Schwarz algorithm tuned to highly heterogeneous media, in Domain Decomposition Methods: Theory and Applications, vol. 25 of Gakuto International Series. Mathematical Sciences Applications, pp. 65-93, Gakkotosho, Tokyo, Japan, 2006.

C. Farhat and P. Le Tallec, Vista in Domain Decomposition Methods," Computer Methods in Applied Mechanics and Engineering, vol. 184, no. 2-4, pp. 143-520, 2000. https://doi.org/10.1016/S0045-7825(99)00226-1

Glowinski, R, Golub, GH, Meurant, GA, Periaux, J: Domain Decomposition Methods for Partial Differential Equations. SIAM, Philadelphia (1988). ISBN:0-89871-220-3.

F. Magoul'es and D. Rixen, Domain decomposition methods :recent advances and new challengesin engineering, Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 8, pp. 1345-1346, 2007. https://doi.org/10.1016/j.cma.2006.09.003

J. Nitsche, L∞-convergence of finite element approximations, in Proceedings of the Symposium on Mathematical Aspects of Finite Element Methods, vol. 606 of Lecture Notes in Mathematics, 261-274, 1977. https://doi.org/10.1007/BFb0064468

S. H. Lui, "On linear monotone iteration and Schwarz methods for nonlinear elliptic PDEs," Numerische Mathematik, vol. 93, no. 1, pp. 109-129, 2002. https://doi.org/10.1007/BF02679439

A. Verfurth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley Teubner, Stuttgart, Germany, 1996.

P. L. Lions. On the Schwarz alternating method. I. First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987). SIAM. Philadelphia, 1988. pp. 1-42.

P. L. Lions. On the Schwarz alternating method. II. Stochastic interpretation and order properties, Domain Decomposition Methods (Los angeles, Calif, 1988). SIAM. Philadelphia 1989. pp 47-70.

F. C. Otto and G. Lube, A posteriori estimates for a non-overlapping domain decomposition method, Computing, vol. 62, no. 1, pp. 27-43, 1999. https://doi.org/10.1007/s006070050011

C. Bernardi, T. Chacon Rebollo, E. Chacon Vera, and D. Franco Coronil, A posteriori error analysis for two overlapping domain decomposition techniques, Applied Numerical Mathematics, vol. 59, no.6, pp. 1214-1236, 2009. https://doi.org/10.1016/j.apnum.2008.06.004

Ph. Cortey-Dumont, On finite element approximation in the L∞-norm of variational inequalities, Numerische Mathematik 47 (1985), no. 1, 45-57. https://doi.org/10.1007/BF01389875

Publiée
2021-12-18
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