A posteriori error analysis for parabolic variational inequalities
Moon, Kyoung-Sook ; Nochetto, Ricardo H. ; Petersdorff, Tobias Von ; Zhang, Chen-Song
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007), p. 485-511 / Harvested from Numdam
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Motivated by the pricing of American options for baskets we consider a parabolic variational inequality in a bounded polyhedral domain Ω⊂ℝ d with a continuous piecewise smooth obstacle. We formulate a fully discrete method by using piecewise linear finite elements in space and the backward Euler method in time. We define an a posteriori error estimator and show that it gives an upper bound for the error in L 2 (0,T;H 1 (Ω)). The error estimator is localized in the sense that the size of the elliptic residual is only relevant in the approximate non-contact region, and the approximability of the obstacle is only relevant in the approximate contact region. We also obtain lower bound results for the space error indicators in the non-contact region, and for the time error estimator. Numerical results for d=1,2 show that the error estimator decays with the same rate as the actual error when the space meshsize h and the time step τ tend to zero. Also, the error indicators capture the correct behavior of the errors in both the contact and the non-contact regions.

Publié le : 2007-01-01
DOI : https://doi.org/10.1051/m2an:2007029
Classification:  58E35,  65N15,  65N30 
@article{M2AN_2007__41_3_485_0,
     author = {Moon, Kyoung-Sook and Nochetto, Ricardo H. and Petersdorff, Tobias Von and Zhang, Chen-Song},
     title = {A posteriori error analysis for parabolic variational inequalities},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {41},
     year = {2007},
     pages = {485-511},
     doi = {10.1051/m2an:2007029},
     mrnumber = {2355709},
     zbl = {pre05289382},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2007__41_3_485_0}
}

Moon, Kyoung-Sook; Nochetto, Ricardo H.; Petersdorff, Tobias Von; Zhang, Chen-Song. A posteriori error analysis for parabolic variational inequalities. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007) pp. 485-511. doi : 10.1051/m2an:2007029. http://gdmltest.u-ga.fr/item/M2AN_2007__41_3_485_0/

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