Elliptic Reconstruction and A Posteriori Error Estimates for Parabolic Variational Inequalities

Elliptic reconstruction property, originally introduced by Makridakis and Nochetto for linear parabolic problems, is a well-known tool to derive optimal a posteriori error estimates. No such results are known for nonlinear and nonsmooth problems such as parabolic variational inequalities (VIs). This article establishes the elliptic reconstruction property for parabolic VIs and derives a posteriori error estimates in $L^{\infty}(0,T;L^{2}(\Omega))$ and $L^{\infty}(0,T;L^{\infty}(\Omega))$, respectively. As an application, the residual-type error estimates are presented.