A posteriori error analysis for parabolic variational inequalities
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 3, p. 485-511

Motivated by the pricing of American options for baskets we consider a parabolic variational inequality in a bounded polyhedral domain $\Omega \subset {ℝ}^{d}$ 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 ${L}^{2}\left(0,T;{H}^{1}\left(\Omega \right)\right)$. 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 $\tau$ tend to zero. Also, the error indicators capture the correct behavior of the errors in both the contact and the non-contact regions.

DOI : https://doi.org/10.1051/m2an:2007029
Classification:  58E35,  65N15,  65N30
Keywords: a posteriori error analysis, finite element method, variational inequality, american option pricing
@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},
publisher = {EDP-Sciences},
volume = {41},
number = {3},
year = {2007},
pages = {485-511},
doi = {10.1051/m2an:2007029},
zbl = {pre05289382},
mrnumber = {2355709},
language = {en},
url = {http://www.numdam.org/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, Volume 41 (2007) no. 3, pp. 485-511. doi : 10.1051/m2an:2007029. http://www.numdam.org/item/M2AN_2007__41_3_485_0/

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