A Posteriori Error Estimates in the Maximum Norm for Parabolic Problems

We derive a posteriori error estimates in the $L_\infty((0,T];L_\infty(\Omega))$ norm for approximations of solutions to linear parabolic equations. Using the elliptic reconstruction technique introduced by Makridakis and Nochetto and heat kernel estimates for linear parabolic problems, we first prove a posteriori bounds in the maximum norm for semidiscrete finite element approximations. We then establish a posteriori bounds for a fully discrete backward Euler finite element approximation. The elliptic reconstruction technique greatly simplifies our development by allowing the straightforward combination of heat kernel estimates with existing elliptic maximum norm error estimators.