A posteriorierror analysis for parabolic variational inequalities

Abstract

Motivated by the pricing of American options for baskets we consider a parabolic varia- tional inequality in a bounded polyhedral domain Ω ⊂ R d with a continuous piecewise smooth obstacle. We formulate a fully discrete method by using piecewise linear finite elements in space and the back- ward Euler method in time. We define an a posteriori error estimator and show that it gives an upper bound for the error in L 2 (0 ,T ; H 1 (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.