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.

翻译：椭圆重构性质最初由Makridakis和Nochetto针对线性抛物问题提出，是推导最优后验误差估计的经典工具。然而，对于非线性、非光滑问题（如抛物变分不等式（VIs）），目前尚无此类结果。本文建立了抛物VIs的椭圆重构性质，并分别推导了$L^{\infty}(0,T;L^{2}(\Omega))$和$L^{\infty}(0,T;L^{\infty}(\Omega))$范数下的后验误差估计。作为应用，给出了残差型误差估计。