An approach is presented to efficiently compute the three-dimensional fluid flow and heat transfer in the melt conveying section of a single-screw extruder. The analysis is based on a moving barrel formulation with an unwound screw channel of rectangular cross section. The viscosity of the polymer melt is described by the power law model together with the exponential temperature dependence. The solution technique consists of marching in the down-channel direction using a fully implicit scheme and solving the resulting equations in each cross-sectional plane by the Galerkin finite element method. Unlike most previous analyses, the present approach takes full account of the recirculatory nature of the cross-channel flow, and therefore also, the cross-stream convection effects are properly accounted for. The present numerical scheme is capable of producing converged wellbehaved solutions without the traditional need for upwinding even at very high values of the Peclet number. The numerical results obtained establish that the cross-channel recirculating flow and associated convection effects play a major role in determining the flow and heat transfer characteristics in a typical rectangular screw channel having large (but finite) aspect ratio.