The healing times for the growth of thin films on patterned substrates are studied using simulations of two discrete models of surface growth: the Family model and the Das Sarma-Tamborenea (DT) model. The healing time, defined as the time at which the characteristics of the growing interface are ``healed″ to those obtained in growth on a flat substrate, is determined via the study of the nearest-neighbor height difference correlation function. Two different initial patterns are considered in this work: a relatively smooth tent-shaped triangular substrate and an atomically rough substrate with singlesite pillars or grooves. We find that the healing time of the Family and DT models on aL x L triangular substrate is proportional to L-z, where z is the dynamical exponent of the models. For the Family model, we also analyze theoretically, using a continuum description based on the linear Edwards-Wilkinson equation, the time evolution of the nearest-neighbor height difference correlation function in this system. The correlation functions obtained from continuum theory and simulation are found to be consistent with each other for the relatively smooth triangular substrate. For substrates with periodic and random distributions of pillars or grooves of varying size, the healing time is found to increase linearly with the height (depth) of pillars (grooves). We show explicitly that the simulation data for the Family model grown on a substrate with pillars or grooves do not agree with results of a calculation based on the continuum Edwards-Wilkinson equation. This result implies that a continuum description does not work when the initial pattern is atomically rough. The observed dependence of the healing time on the substrate size and the initial height (depth) of pillars (grooves) can be understood from the details of the diffusion rule of the atomistic model. The healing time of both models for pillars is larger than that for grooves with depth equal to the height of the pillars. The calculated healing time for both Family and DT models is found to depend on how the pillars and grooves are distributed over the substrate. (C) 2014 Elsevier B.V. All rights reserved.