In this paper, we study the numerical analysis and solution of the classical Stokes problem that models the steady state incompressible fluid flows. The study is based on the mixed finite element discretization of the problem domain. The heart of the paper is to present the preconditioned conjugate gradient accelerated Uzawa method applied to solve the resulting indefinite systems of linear algebraic equations from the mixed finite element discretization. We investigate the classical Uzawa and the preconditioned Uzawa method and present comparative results on the performance of these iterative schemes in terms of computational time and iterative counts. The results show that the preconditioned conjugate gradient is effective in accelerating the performance of the inexact Uzawa. We also present the main theoretical convergence results. We study the problem in a two dimensional setting using the Hood-Taylor  pair of finite elements. The incompressible flow iterative solution software (IFISS) matlab toolbox is used to assemble the matrices. We present the numerical results to illustrate the efficiency and robustness of the preconditioned conjugate gradient Uzawa scheme.

classical Stokes problem, mixed finite element method, classical Uzawa, inexact Uzawa method.