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A posteriori error analysis for parabolic variational inequalities

Published online by Cambridge University Press:  02 August 2007

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 we consider a parabolic variational inequality in a bounded polyhedral domain $\Omega\subset\mathbb{R}^d$ 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 L2(0,T;H1(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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2007

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