This paper provides a theoretical justification to a overlapping domain decomposition method applied to the solution of time-dependent convection-diffusion problems. The method is based on the partial upwind finite element scheme and the discrete strong maximum principle for steady problem. An error estimate in L∞ (0, T; L∞ (Ω)) is obtained and the fact that convergence factor ρ(т, h) → 0 exponentially as т, h → 0 is also proved under some usual conditions.