Motivated by the pricing of American options for baskets we consider a parabolic variational inequality in a bounded polyhedral domain 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 . 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 show that the error estimator decays with the same rate as the actual error when the space meshsize and the time step tend to zero. Also, the error indicators capture the correct behavior of the errors in both the contact and the non-contact regions.
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: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {485--511}, publisher = {EDP-Sciences}, volume = {41}, number = {3}, year = {2007}, doi = {10.1051/m2an:2007029}, mrnumber = {2355709}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an:2007029/} }
TY - JOUR AU - Moon, Kyoung-Sook AU - Nochetto, Ricardo H. AU - Petersdorff, Tobias Von AU - Zhang, Chen-Song TI - A posteriori error analysis for parabolic variational inequalities JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2007 SP - 485 EP - 511 VL - 41 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an:2007029/ DO - 10.1051/m2an:2007029 LA - en ID - M2AN_2007__41_3_485_0 ER -
%0 Journal Article %A Moon, Kyoung-Sook %A Nochetto, Ricardo H. %A Petersdorff, Tobias Von %A Zhang, Chen-Song %T A posteriori error analysis for parabolic variational inequalities %J ESAIM: Modélisation mathématique et analyse numérique %D 2007 %P 485-511 %V 41 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an:2007029/ %R 10.1051/m2an:2007029 %G en %F 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: Modélisation mathématique et analyse numérique, Volume 41 (2007) no. 3, pp. 485-511. doi : 10.1051/m2an:2007029. http://archive.numdam.org/articles/10.1051/m2an:2007029/
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