A posteriori error analysis for parabolic variational inequalities

¹ Department of Mathematics and Information, Kyungwon University, Bokjeong-dong, Sujeong-gu, Seongnam-si, Gyeonggi-do, 461-701, Korea. ksmoon@kyungwon.ac.kr
² Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA. rhn@math.umd.edu
³ Department of Mathematics, University of Maryland, College Park, MD 20742, USA. tvp@math.umd.edu; zhangcs@math.umd.edu

Received: 20 January 2006

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.