We consider the steady Navier-Stokes problem in two dimensions to be
solved using numerical iterative methods. Firstly, we consider a test
problem with a known solution and therefore the results of the numerical
solutions can be compared with the exact solution. Secondly, we solve the
lid-driven cavity problem at high Reynolds number, and the results can be
compared with previous results. We discretize the problem using Chebyshev
discretization in the 
and 
directions. Because of the
non-linearity of the Navier-Stokes equations, Newton linearization is
used to work in terms of correction terms. Here, we combine the use of
iterative Newton linearization with a direct solver to solve the
vorticity-streamfunction formulation. The direct solver employs the back
substitution Gauss elimination method with piovting for solving the
linear system. We solve the Navier-Stokes problem exploiting the accuracy
of the use of Chebyshev discretization to obtain the numerical solution
at high Reynolds number and to recognize the type of resulting matrix.
The resulting matrix which arise from the discretization is usually full
and the linear system is solved by a direct solver. In the second half of
the work, we try to solve the linear system araised from the
discretization using the iterative method (GMRES) at high Reynolds number
and compare it with the direct solver. If the time help, a preconditioner
will be used to improve the ierations. At high Reynolds number, GMRES
diverges while the direct solver can do higher than the GMRES for the
same discretisation grid. To conclude, a relation between the direct
solver and the GMRES can be drawn. Also, a relation between the Reynolds
number and convergence of GMRES can be observed.