Elliptic Reconstruction and A Posteriori Error Estimates for Parabolic Variational Inequalities

TL;DR

This work extends the elliptic reconstruction methodology to nonlinear and nonsmooth parabolic variational inequalities by introducing a reconstruction map $\mathcal{W}(t)$ that incorporates the discrete Lagrange multiplier $\sigma_h$. A key inequality links the reconstruction error to computable residuals, enabling robust a posteriori error estimates in the $L^{\infty}(0,T;L^{2}(\Omega))$-norm for semidiscrete finite element approximations. The authors derive abstract energy-norm error bounds and then specialize to residual-type estimators $\eta^0_h$ and $\eta^1_h$, with distinct formulations for $k\ge 2$ and $k=1$, providing practical guidelines for adaptive methods. Overall, the results offer a rigorous, computable framework for error control and adaptivity in nonlinear parabolic VIs with obstacles, significantly advancing numerical analysis in this challenging class of problems.

Abstract

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 $L^{\infty}(0,T;L^{2}(Ω))$. The estimator consists of discrete complementarity terms and standard residual. As an application, the residual-type error estimates are presented.

Theorems & Definitions (18)

• Remark 2.1
• Remark 2.2
• Lemma 3.1
• Lemma 4.1
• proof
• Lemma 5.1
• proof
• Lemma 5.2
• Remark 5.3
• proof