The Elrod algorithm is commonly used to analyze the cavitation problems in hydrodynamic lubrication films. The successive over relaxation (SOR) method is a natural choice for speeding up the Elrod algorithm. However, a direct application of the SOR method can lead to poor efficiency due to the mixed elliptic-hyperbolic properties contributed by the mass-conserving Reynolds equation. This paper presents a practical technique for obtaining dynamic relaxation factors that accelerate the SOR-based Elrod algorithm. This technique is based on the power method, which allows the automatic calculation of dynamic optimal relaxation factors during free boundary movements without prior trials. The preconditioning technique is presented that takes advantage of switch functions to avoid the negative effect of the hyperbolic-dominated region. The cases of a journal bearing and a thrust bearing were analyzed. It shows that the new method significantly reduces the time consumed in finding the optimal relaxation factor. The solution efficiency of this method is higher than that of the traditional Elrod algorithm using a constant, optimal relaxation factor. This improvement varies depending on the specific problems; for the thrust bearing case, the solution time is reduced by approximately 43%. The proposed method is practical because it retains the advantage of a matrix-free implementation.