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A posteriori error analysis for parabolicvariational inequalities

Published online by Cambridge University Press:  02 August 2007

Kyoung-Sook Moon ,
Ricardo H. Nochetto ,
Tobias von Petersdorff  and
Chen-song Zhang
Kyoung-Sook Moon
Affiliation:
Department of Mathematics and Information, Kyungwon University, Bokjeong-dong, Sujeong-gu, Seongnam-si, Gyeonggi-do, 461-701, Korea. ksmoon@kyungwon.ac.kr
Ricardo H. Nochetto
Affiliation:
Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA. rhn@math.umd.edu
Tobias von Petersdorff
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA. tvp@math.umd.edu; zhangcs@math.umd.edu
Chen-song Zhang
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA. tvp@math.umd.edu; zhangcs@math.umd.edu
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Abstract

Motivated by the pricing of American options for baskets weconsider a parabolic variational inequality in a boundedpolyhedral domain $\Omega\subset\mathbb{R}^d$ with a continuous piecewisesmooth obstacle. We formulate a fully discrete method by usingpiecewise linear finite elements in space and the backward Eulermethod in time. We define an a posteriori error estimator and showthat it gives an upper bound for the error inL2(0,T;H1 (Ω)). The error estimator is localized in thesense that the size of the elliptic residual is only relevant inthe approximate non-contact region, and the approximability of theobstacle is only relevant in the approximate contact region. Wealso obtain lower bound results for the space error indicators inthe non-contact region, and for the time error estimator.Numerical results for d=1,2 show that the error estimator decayswith the same rate as the actual error when the space meshsize hand the time step τ tend to zero. Also, the error indicatorscapture the correct behavior of the errors in both the contact andthe non-contact regions.

Keywords

A posteriori error analysisfinite element methodvariational inequalityAmerican option pricing.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 3 , May 2007 , pp. 485 - 511
Copyright
© EDP Sciences, SMAI, 2007

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