  The aim of this work is the numerical homogenization of a parabolic problem
with several time and spatial scales using the heterogeneous multiscale method.
We replace the actual cell problem with an alternate one, using Dirichlet
boundary and initial values instead of periodic boundary and time conditions.
Further, we give a detailed a priori error analysis of the fully discretized,
i.e., in space and time for both the macroscopic and the cell problem, method.
Numerical experiments illustrate the theoretical convergence rates.

Numerical Analysis;Numerical Analysis