We derive a posteriori error estimates in the$L_\infty((0,T];L_\infty(\Omega))$ norm for approximations of solutions tolinear para bolic equations. Using the elliptic reconstruction techniqueintroduced by Makridakis and Nochetto and heat kernel estimates for linearparabolic pr oblems, we first prove a posteriori bounds in the maximum norm forsemidiscrete finite element approximations. We then establish a posterioribounds for a fully discrete backward Euler finite element approximation. Theelliptic reconstruction technique greatly simplifies our development by allow\ing the straightforward combination of heat kernel estimates with existingelliptic maximum norm error estimators.
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