Solution Method
In block form equation equations 6.1 and 6.2 takes the form of a saddle point problem
![$\displaystyle \left[ \begin{array}{cc} A & B^{*} \\ B & 0 \\ \end{array} \right...
...p \\ \end{array} \right] =\left[ \begin{array}{c} G \\ 0 \\ \end{array} \right]$](img771.png) |
(94) |
where
is coercive, self-adjoint linear operator in a suitable Hilbert space,
is the
divergence operator and
is it adjoint operator (=gradient operator). For more details on the mathematics see references [3,4].
We use iterative techniques to solve this problem. To make sure that the incomressibilty condition holds
with sufficient accuracy we check for
 |
(95) |
where
is the desired relative accuracy and
 |
(96) |
defines the
-norm. We use the Uzawa scheme to solve the problem.
In fact the first equation in 6.5 gives for a known pressure
 |
(97) |
which is inserted into the second equation leading to
 |
(98) |
with the Schur complement
. This problem can be solved iteratively
with the preconditioner
defined as
by solving
 |
(99) |
see [8] for more details. Note that the residual for the current approximation
is given as
 |
(100) |
where
is given by 6.8.
If one uses the generalized minimal residual method (GMRES)
the method is directly applied to the preconditioned system
 |
(101) |
We use the norm
 |
(102) |
Notice that for the residual
one has
 |
(103) |
If
provides an initial guess for the pressure we use 6.8 to get a first initial guess for the
velocity
which we use to set an absolute tolerance
.
The GMRES is terminated when
 |
(104) |
Notice that
so we we can expect that
the target stopping criterion 6.6 is fullfilled. However, if
is very different from the
initial choice of
the value of
is corrected and GMRES is restarted with a new tolerance. For time dependend problems this apprach works well as value for
form a previous time step provides a good initial guess.
Alternatively, as
is symmetric and positive definite one can apply the preconditioned conjugate gradient method (PCG) . PCG use the norm
 |
(105) |
To take the extra factor
into consideration when checking the stopping criterion we use the following
definition for
:
 |
(106) |
esys@esscc.uq.edu.au