Asymptotic behavior and a posteriori error estimates in Sobolev space for the generalized overlapping domain decomposition method for evolutionary HJB equation with nonlinear source terms. Part 1
-
1723
Downloads
-
2564
Views
Authors
Salah Boulaaras
- Department Of Mathematics, College Of Sciences and Arts, Al-Qassim University, Al-Rass, Kingdom Of Saudi Arabia.
Abstract
A posteriori error estimates for the generalized overlapping domain decomposition method with Dirichlet
boundary conditions on the boundaries for the discrete solutions on subdomains of evolutionary HJB equation
with nonlinear source terms are established using the semi-implicit time scheme combined with a FInite
element spatial approximation. Also the techniques of the residual a posteriori error analysis are used. Moreover,
using Benssoussan-Lions' algorithm, an asymptotic behavior in \(H^1_0\)-norm is deduced. Furthermore,
the results of some numerical experiments are presented to support the theory.
Share and Cite
ISRP Style
Salah Boulaaras, Asymptotic behavior and a posteriori error estimates in Sobolev space for the generalized overlapping domain decomposition method for evolutionary HJB equation with nonlinear source terms. Part 1, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 3, 736--756
AMA Style
Boulaaras Salah, Asymptotic behavior and a posteriori error estimates in Sobolev space for the generalized overlapping domain decomposition method for evolutionary HJB equation with nonlinear source terms. Part 1. J. Nonlinear Sci. Appl. (2016); 9(3):736--756
Chicago/Turabian Style
Boulaaras, Salah. "Asymptotic behavior and a posteriori error estimates in Sobolev space for the generalized overlapping domain decomposition method for evolutionary HJB equation with nonlinear source terms. Part 1." Journal of Nonlinear Sciences and Applications, 9, no. 3 (2016): 736--756
Keywords
- A posteriori error estimates
- GODDM
- Dirichlet boundary conditions
- algorithm
- asymptotic behavior.
MSC
References
-
[1]
M. Ainsworth, J. T. Oden, A posteriori error estimation in finite element analysis, Wiley-Interscience [John Wiley & Sons], New York (2000)
-
[2]
H. Benlarbi, A. S. Chibi , A posteriori error estimates for the generalized overlapping domain decomposition methods, J. Appl. Math., 2012 (2012), 15 pages.
-
[3]
A. Bensoussan, J. L. Lions, Contrôle impulsionnel et in équations quasi-variationnelles, Gauthier-Villars, California (1984)
-
[4]
C. Bernardi, T. Chacon Rebollo, E. Chacon Vera, D. Franco Coronil , A posteriori error analysis for two-overlapping domain decomposition techniques, Appl. Numer. Math., 59 (2009), 1214-1236.
-
[5]
S. Boulaaras, K. Habita, M. Haiour, Asymptotic behavior and a posteriori error estimates for the generalized overlapping domain decomposition method for parabolic equation, Bound. Value Probl., 2015 (2015), 16 pages.
-
[6]
S. Boulaaras, M. Haiour, The maximum norm analysis of an overlapping Shwarz method for parabolic quasi-variational inequalities related to impulse control problem with the mixed boundary conditions, Appl. Math. Inf. Sci., 7 (2013), 343-353.
-
[7]
S. Boulaaras, M. Haiour, The finite element approximation of evolutionary Hamilton-Jacobi-Bellman equations with non-linear source terms, Indag. Math., 24 (2013), 161-173.
-
[8]
S. Boulaaras, M. Haiour , A new proof for the existence and uniqueness of the discrete evolutionary HJB equation, Appl. Math. Comput., 262 (2015), 42-55.
-
[9]
S. Boulaaras, M. Haiour , A general case for the maximum norm analysis of an overlapping Schwarz methods of evolutionary HJB equation with nonlinear source terms with the mixed boundary conditions , Appl. Math. Inf. Sci., 9 (2015), 1247-1257.
-
[10]
M. Boulbrachene, M. Haiour , The finite element approximation oft Hamilton-Jacobi-Bellman equations, Comput. Mah. Appl., 41 (2001), 993-1007.
-
[11]
T. F. Chan, T. Y. Hou, P. L. Lions, Geometry related convergence results for domain decomposition algorithms, SIAM J. Numer. Anal., 28 (1991), 378-391.
-
[12]
P. G. Ciarlet, P. A. Raviart, Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Engrg., 2 (1973), 17-31.
-
[13]
P. Cortey-Dumont , Approximation numerique d une inequation quasi-variationnelle liee a des problemes de gestion de stock, RAIRO Anal. Numer., 14 (1980), 335-346.
-
[14]
P. Cortey-Dumont, On finite element approximation in the \(L^\infty\)-norm of variational inequalities, Numer. Math., 47 (1985), 45-57.
-
[15]
J. Douglas, C. S. Huang, An accelerated domain decomposition procedure based on Robin transmission conditions, BIT, 37 (1997), 678-686.
-
[16]
B. Engquist, H. K. Zhao, Absorbing boundary conditions for domain decomposition, Appl. Numer. Math., 27 (1998), 341-365.
-
[17]
C. Farhat, P. Le Tallec, Vista in domain decomposition methods , Comput. Methods Appl. Mech. Eng., 184 (2000), 143-520.
-
[18]
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., 121 (2011), 481-493.
-
[19]
P. L. Lions , On the Schwarz alternating method. I. First international symposium on domain decomposition methods for partial differential equations, SIAM, Philadelphia, (1988), 1-42.
-
[20]
P. L. Lions , On the Schwarz alternating method. II.Stochastic interpretation and order properties. domain decomposition methods, SIAM, Philadelphia, (1989), 47-70.
-
[21]
Y. Maday, F. Magoules, Improved ad hoc interface conditions for Schwarz solution procedure tuned to highly heterogeneous media, Appl. Math. Model., 30 (2006), 731-743.
-
[22]
Y. Maday, F. Magoules , A survey of various absorbing interface conditions for the Schwarz algorithm tuned to highly heterogeneous media, in domain decomposition methods, Gakuto international series, Math. Sci. Appl., 25 (2006), 65-93.
-
[23]
F. Nataf , Recent developments on optimized Schwarz methods, Lect. Notes Comput. Sci. Eng., Springer, Berlin, (2007), 115-125.
-
[24]
F. C. Otto, G. Lube, A posteriori estimates for a non-overlapping domain decomposition method, Computing, 62 (1999), 27-43.
-
[25]
A. Quarteroni, A. Valli , Domain decomposition methods for partial differential equations, The Clarendon Press, Oxford University Press, New York (1999)
-
[26]
D. Rixen, F. Magoules, Domain decomposition methods: recent advances and new challenges in engineering, Comput. Methods Appl. Mech. Engrg., 196 (2007), 1345-1346.
-
[27]
A. Toselli, O. Widlund , Domain decomposition methods algorithms and theory, Springer, Berlin (2005)
-
[28]
A. Verurth, A review of a posteriori error estimation and adaptive mesh-refinement techniques, Wiley-Teubner, Stuttgart (1996)