Using spectral results for Schur complement operators we prove a convergence result for the inexact Uzawa algorithm on general Hilbert spaces. We prove that for any symmetric and coercive saddle point problem, the inexact Uzawa algorithm converges, provided that the inexact process for inverting the residual at each step has the relative error smaller than a computable fixed threshold. As a consequence, we provide a new type of algorithms for discretizing saddle point problems, which implement the inexact Uzawa algorithm at the continuous level as a multilevel algorithm. The discrete stability Ladyshenskaya-Babu\v{s}ca-Brezzi (LBB) condition might not be satisfied. The convergence result for the algorithm at the continuous level, combined with standard techniques of discretization and a posteriori error estimates leads to new and efficient algorithms for solving saddle point systems. Numerical results supporting the efficiency of the algorithm are presented for the Stokes Equations and for the div-curl systems