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Using nonequilibrium molecular dynamics simulations, we compute the shear viscosity, ηs, of a glass forming polymer melt at temperatures ranging from the normal liquid state down to the supercooled state. For this purpose, the polymer melt is confined between two solid walls and a constant force pointing in direction parallel to the walls is applied on each monomer thus giving rise to a Poiseuille flow. It is shown that ηs(T) does not exhibit an Arrhenius-type behavior but can be described both by a power law (mode coupling theory) and by a Vogel–Fulcher–Tammann law. A similar behavior is observed in recent experiments above the glass transition temperature. The diffusion coefficient is computed using the mean square displacements in direction perpendicular to the flow. Combined with the knowledge of ηs(T), it is then shown that the Stokes–Einstein relation is valid at high temperatures, whereas deviations are observed in the supercooled regime in agreement with experiments. Moreover, the local viscosity, η(z), is also computed and its reliability is discussed. Using the sharp rise of η(z) close to the wall, we estimate zwall, the effective position of the wall. It is found that zwall moves towards the film center at lower T thus leading to a decrease of the (hydrodynamic) width of the system. Furthermore, we observe that the curves for η(z)/ηs at various temperatures superimpose if the data are depicted versus z−zwall(T). This suggests that the spatial and temperature dependence of the local viscosity separate if the effective position of the wall is chosen as a new reference plane.