Research Article

A posteriori error analysis for parabolic variational inequalities

Moon, Kyoung-Sook^a1, Nochetto, Ricardo H.^a2, von Petersdorff, Tobias^a3 and Zhang, Chen-song^a3

^a1 Department of Mathematics and Information, Kyungwon University, Bokjeong-dong, Sujeong-gu, Seongnam-si, Gyeonggi-do, 461-701, Korea. ksmoon@kyungwon.ac.kr

^a2 Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA. rhn@math.umd.edu

^a3 Department of Mathematics, University of Maryland, College Park, MD 20742, USA. tvp@math.umd.edu; zhangcs@math.umd.edu

Abstract

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.

(Received January 20 2006)

(Online publication August 2 2007)

Key Words:

• A posteriori error analysis;
• finite element method;
• variational inequality;
• American option pricing.

Mathematics Subject Classification:

• 58E35;
• 65N15;
• 65N30