In this work, we provide a new analysis of the convergence of the ADMM (Alternating Direction Method of Multipliers) based on the equivalence between the ADMM and the PDHG (Primal-Dual Hybrid Gradient) with overrelaxation. The convergence study of the latter in the smooth case (where the objective function is decomposed in a strongly convex part and a differentiable part with a Lipschitz continuous gradient) allows us to deduce convergence results on both the ADMM and an accelerated variant of the ADMM. Numerical comparisons with the PDHG method and the well-known FISTA are shown in practical cases.