AN ASYMPTOTIC BEHAVIOR AND A POSTERIORI ERROR ESTIMATES FOR THE GENERALIZED SCHWARTZ METHOD OF ADVECTION-DIFFUSION EQUATION

doi:10.1016/S0252-9602(18)30810-5

Abstract

Abstract:

In this paper, a posteriori error estimates for the generalized Schwartz method with Dirichlet boundary conditions on the interfaces for advection-diffusion equation with second order boundary value problems are proved by using the Euler time scheme combined with Galerkin spatial method. Furthermore, an asymptotic behavior in Sobolev norm is deduced using Benssoussan-Lions' algorithm. Finally, the results of some numerical experiments are presented to support the theory.

Key words: a posteriori error estimates, GODDM, advection-diffusion, Galerkin method, Benssoussan-Lions'algorithm

Salah BOULAARAS, Mohammed Said TOUATI, BRAHIM Smail BOUZENADA, Abderrahmane ZARAI. AN ASYMPTOTIC BEHAVIOR AND A POSTERIORI ERROR ESTIMATES FOR THE GENERALIZED SCHWARTZ METHOD OF ADVECTION-DIFFUSION EQUATION[J].Acta mathematica scientia,Series B, 2018, 38(4): 1227-1244.

Trendmd

References

[1] Allahem A, Boulaaras S, Cherif B. A posteriori error estimate of the theta time scheme combined with a finite element spatial approximation for evolutionary HJB equation with linear source terms. Journal of Computational and Theoretical Nanoscience, 2017, 14(2):935-946
[2] Badea L. On the schwarz alternating method with more than two subdomains for monotone problems. SIAM Journal on Numerical Analysis, 1991, 28:179-204
[3] Bensoussan A, Lions J L. Contrôle impulsionnel et In équations Quasi-variationnelles. Dunod, 1982
[4] Boulaaras S, Haiour M. Overlapping domain decomposition methods for elliptic quasi-variational inequalities related to impulse control problem with mixed boundary conditions. Proc Indian Acad Sci (Math Sci), 2011, 121(4):481-493
[5] Boulaaras S, Habita K, Haiour M. Asymptotic behavior and a posteriori error estimates for the generalized overlapping domain decomposition method for parabolic equation. Boundary Value Problems, 2015, 2015:124. DOI:10.1186/s13661-015-0398-1
[6] Boulaaras S. Asymptotic behavior and a posteriori error estimates in Sobolev spaces for the generalized overlapping domain decomposition method for evolutionary HJB equation with non linear source terms, part 1. J Nonlinear Sci Appl, 2016, 9(3):736-756
[7] Boulaaras S, Allahem A, Haiour M, Zennir K, Ghanem S, Cherif B. A posteriori error estimates in H¹(Ω) space for parabolic quasi-variational inequalities with linear source terms related to American options problem. Appl Math Inf Sci, 2016, 10(3):1-14
[8] Douglas Jr J, Huang C S. An accelerated domain decomposition procedure based on Robin transmissionconditions. BIT, Numerical Mathematics, 1997, 37:678-686
[9] Engquist B, Zhao H K. Absorbing boundary conditions for domain decomposition. Applied Numerical Mathematics, 1998, 27:341-365
[10] Kuznetsov YU A. Overlapping domain decomposition method for parabolic problem. Contempary Mathematics, 1994, 157:63-69
[11] Lions P L. On the Schwarz alternating method I//Glowinski R, Golub G H, Meurant G A, Periaux J, eds. First International Symposium on Domain Decomposition Methods for Partial Differential Equations. SIAM, 1988:1-42
[12] Chan T F, Hou T Y, Lions P L. Geometry related convergence results for domain decomposition algorithms. SIAM Journal on Numerical Analysis, 1991, 28:378-391
[13] Quarteroni A, Valli A. Domain Decomposition Methods for Partial Differential Equations. Oxford, UK:The Clarend on Press, 1999
[14] Toselli A, Widlund O. Domain Decomposition Methods Algorithms and Theory. Vol 34 of Springer Seriesin Computational Mathematics. Berlin:Springer, 2005
[15] Maday Y, Magoul'es F. Improved ad hoc interface conditions for Schwarz solution procedure tuned to highly heterogeneous media. Applied Mathematical Modelling, 2006, 30(8):731-743
[16] Maday Y, Magoules F. A survey of various absorbing interface conditions for the Schwarz algorithm tuned to highly heterogeneous media//Domain Decomposition Methods:Theory and Applications. Vol 25 of Gakuto International Series. Mathematical Sciences Applications. Tokyo:Gakkotosho, 2006:65-93
[17] Farhat C, Le Tallec P. Vista in domain decomposition methods. Computer Methods in Applied Mechanics and Engineering, 2000, 184(2/4):143-520
[18] Magoules F, Rixen D. Domain decomposition methods:recent advances and new challengesin engineering. Computer Methods in Applied Mechanics and Engineering, 2007, 196(8):1345-1346
[19] Nataf F. Recent developments on optimized Schwarz methods//Domain Decomposition Methods in Science and Engineering XVI. Vol 55 of Lecture Notes in Computational Science and Engineering. Berlin:Springer, 2007:115-125
[20] Ainsworth M, Oden J T. A Posteriori Error Estimation in Finite Element Analysis. New York:Wiley Interscience, 2000
[21] Verfurth A. A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Stuttgart:Wiley Teubner, 1996
[22] Lions P L. On the Schwarz alternating method I//First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987). Philadelphia:SIAM, 1988:1-42
[23] Lions P L. On the Schwarz alternating method Ⅱ//Stochastic Interpretation and Order Properties, Domain Decomposition Methods (Los angeles, Calif, 1988). Philadelphia:SIAM, 1989:47-70
[24] Lube G, Muller L, Otto F C. A non-overlapping domain decomposition method for advection-diffusion problem. Computing, 2000, 64(1):49-68
[25] Otto F C, Lube G. A posteriori estimates for a non-overlapping domain decomposition method. Computing, 1999, 62(1):27-43
[26] Bernardi C, Chacon Rebollo T, Chacon Vera E, Franco Coronil D. A posteriori error analysis for twooverlapping domain decomposition techniques. Applied Numerical Mathematics, 2009, 59(6):1214-1236
[27] Benlarbi H, Chibi A S. A posteriori error estimates for the generalized overlapping domain decomposition methods. Journal of Applied Mathematics, 2012, Article ID 947085