The Elrod-Adams cavitation model is a commonly used model in the theory of lubricated contacts. Two new interpretations of the model are introduced, a fixed point and an optimal control problem, based on an analytical liquid ratio representation and a suitable cost functional, respectively. For both, possible discretizations are proposed. A rigorous convergence analysis, crucial to ensure convergence of solutions of the discretized models to the solution of the continuous Elrod-Adams model, is possible. Two solution algorithms using a finite element method are presented and applied to two numerical journal bearing experiments. A comparison with the Fischer-Burmeister-Newton-Schur algorithm shows a lower cost per iteration and partly a higher accuracy. Depending on the problem, the overall computational cost is also lower.