Abstract
The smoothing property of parabolic equations is used to show that for times uniformly bounded away from the initial time the continuous-in-time Galerkin approximation and the Galerkin approximation based on the Crank–Nicolson time discretization give optimal order errors in the maximum norm under the mild restriction that the initial data for the Galerkin processes are optimally close to initial data for the parabolic boundary value problem in the mean square sense.
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