A posteriori error analysis for parabolic variational inequalities
Department of Mathematics and Information,
Kyungwon University, Bokjeong-dong, Sujeong-gu, Seongnam-si, Gyeonggi-do, 461-701, Korea. firstname.lastname@example.org
2 Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA. email@example.com
3 Department of Mathematics, University of Maryland, College Park, MD 20742, USA. firstname.lastname@example.org; email@example.com
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 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.
Mathematics Subject Classification: 58E35 / 65N15 / 65N30
Key words: A posteriori error analysis / finite element method / variational inequality / American option pricing.
© EDP Sciences, SMAI, 2007