An essential difficulty in the solution of the coupled momentum equations and continuity equation (8.1), (8.2) or its (time discretized form (for example (8.8), (8.9), is the absence of the pressure in the continuity equation. If we consider the system of equations as one large system of linear equations to be solved, this means that in the part corresponding to the continuity equations we have zeros at the main diagonal. Formally equations (8.8), (8.9) may be written as:
where 
is only non-zero if non-zero Dirichlet
boundary conditions for the velocity are prescribed.
The solution of systems of equations of the form (9.1) is in
general more difficult for a linear solver than the solution of equations
arising from the discretization of standard convection-diffusion equations.
There are several ways to solve this problem. One of the possible ways is
to perturb the continuity equation. This leads to methods like the penalty
method or Uzawa iterations. An alternative way to solve the problem is
formed by projection methods. In these methods first the pressure at the
new level is estimated, for example by the old pressure, and then the
momentum equations are solved yielding an intermediate velocity field. By
projecting this velocity onto the space of divergence-free vector fields a
new velocity and pressure may be computed. An important representant of
this class is the so-called pressure-correction method, which will be
treated in 9.2.
