Hello everyone,
I am currently solving a stabilized version of the Navier-Stokes equation 
using dealii and everything is working well when I use a direct solver.
I would like (obviously?) to extend it to use a FGMRES iterative solver to 
be able to tackle very large systems. I have ported those aspects to use 
Trilinos.

The weak form  is (for now we can neglect the temperature related term) :


Concretely, the additional term due to the stabilization affect the 
velocity block in a SUPG manner and the pressure block by adding an 
element-dependent stiffness matrix.
I have read that ILU and ILUT type of preconditioners are the best-suited  
for this type of systems, but I find myself unable to orient my choice of 
parameters.
When we look at the constructor for the AdditionalData of ILU and ILUT 
preconditioner there are the following parameters:
ILU :
      AdditionalData(const unsigned int ilu_fill = 0,
                     const double       ilu_atol = 0.,
                     const double       ilu_rtol = 1.,
                     const unsigned int overlap  = 0);
ILUT:
      AdditionalData(const double       ilut_drop = 0.,
                     const unsigned int ilut_fill = 0,
                     const double       ilut_atol = 0.,
                     const double       ilut_rtol = 1.,
                     const unsigned int overlap   = 0);

The documentation suggest an atol between 1e-2 and 1e-5 and a rtol of the 
order of 1.01. However, what should my ilu_fill and ilut_drop be? I 
understand that increasing the fill will increase the memory consumption 
and that I should try to reach a compromise, but right now I find that it's 
mostly the speed of my iterative solve that is the problem. Are there any 
guidelines to orient my choice of parameters? Am I going with the right 
type of pre-conditioner or am I in the wrong here?
Thank you very much for your help.

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