A posteriori error analysis for parabolic variational inequalities

Moon, Kyoung-Sook; Nochetto, Ricardo H.; Petersdorff, Tobias Von; Zhang, Chen-Song

ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 3, pp. 485-511.

Motivated by the pricing of American options for baskets we consider a parabolic variational inequality in a bounded polyhedral domain $\Omega \subset {ℝ}^d$ 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 $L^2(0,T;H^1\left(\Omega \right))$. 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.

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