Chensong Zhang
University of Maryland

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 $ L^2(0,T;H^1(\Omega))$. 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 $ \tau$ tend to zero. Also, the error indicators capture the correct behavior of the errors in both the contact and the non-contact regions.