Solution Method
In practical applications it is an advantage to calculate the pressure
as a correction of a 'static' pressure
which is the solution of
With setting
and
and
 |
(111) |
we can assume that
and
.
We set
on  |
(112) |
and
and define the operator
defined by
 |
(114) |
and the operator
defined by
 |
(115) |
In operator notation the Darcy problem 6.18 is written in the form
 |
(116) |
We solve this equation by minimising the functional
 |
(117) |
over
where
denotes the norm in
. A simple calculation shows that
one has to solve
 |
(118) |
for all
and
.which translates back into operator notation
 |
(119) |
where
and
denote the adjoint operators.
In [19] it has been shown that this problem is continuous and coercive in
and therefore has a unique solution. Also standart FEM methods can be used for discretization. It is also possible
to solve the problem is coupled form, however this approach leads in some cases to a very ill-conditioned stiffness matrix in particular in the case of a very small or large permability (
or
)
The approach we are taking is to eliminate the velocity
from the problem. Assuming that
is known we have
 |
(120) |
(notice that
is coercive in
) which is inserted into the second equation
 |
(121) |
which is
 |
(122) |
We use the PCG method to solve this. The residual
(
) is given as
 |
(123) |
So in a partical implementation we use
to represent the residual.
The evaluation of the iteration operator for a given
is then
returning
where
is the solution of
 |
(124) |
We use
as a preconditioner for the iteration operator
. So the application of the preconditioner to
representing the residual is given by solving
implemented by solving
 |
(125) |
The residual norm used in the PCG is given as
 |
(126) |
The iteration is terminated if
ATOL |
(127) |
where we set
where rtol is a given relative tolerance and
atol is a given absolute tolerance (typically
).
Notice that if
and
both are zero, the pair
is a solution.
The problem is that ATOL is depending on the solution
(and
calculated form 6.31). In partcice one use the initial guess for
to get a first value for ATOL. If the stopping crierion is met in the PCG iteration, a new
is calculated from the current pressure approximation and ATOL is recalculated. If 6.38 is still fullfilled the calculation is terminated and
is returned. Otherwise PCG is restarted with a new ATOL.
esys@esscc.uq.edu.au