By Constantin Bacuta.
Abstract: Based on 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 $1/3$. As a consequence, we provide 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 (LBB) condition might not be satisfied. Numerical results supporting the efficiency of the algorithm are presented for the Stokes Equations.