[petsc-users] Iterative solver and condition number from FDM + fill-in

[Jed Brown]

"Appel, Thibaut" <t.appel17 at imperial.ac.uk> writes:

> Dear users,
>
> I’m having trouble finding a PC/KSP pair that works for my problem in parallel.
> I’m solving linearized Navier-Stokes PDE’s discretized with a finite difference method in 2D or 3D in a logically rectangular grid, in complex arithmetic.

Compressible or incompressible?  Staggered or centered grid?  Why complex arithmetic?

> It obviously works fine with a direct solver but also with GMRES + ILU(3) in sequential.
>
> I tried different combinations such as
> -ksp_type gmres -pc_type asm -sub_pc_type ilu
> -ksp_type gmres -pc_type bjacobi -sub_pc_type ilu
> 	
> but cannot get the relative residuals below 10^(-2), after 2,000 iterations - even with increasing the number of ILU fill-in levels (up to 5), the number of GMRES restarts (300 to 1000), options such as -ksp_initial_guess_nonzero or -ksp_gmres_cgs_refinement_type refine_always. -ksp_monitor_true_residual does not seem to give more information?
> Maybe there’s room for more experimentation but if you could suggest a way to have a better diagnostic? 
>
> With the different equation sets I’m working with, the condition numbers estimated with the petsc faq method vary between 10^3 and 10^7.
> On top of that I have ridiculous fill-in and have to set -pc_factor_fill to 14, up to 35 (!) sometimes.
>
> For our application we need a lot of discretization points in one spatial direction and I read somewhere that condition number scales with the square of discretization steps for FD methods. But is there a way to reduce it in my case?
> I’m also aware that fill-in should be inevitably expected when you have a sparse matrix with a banded structure arising from a FDM. But I was wondering if there’s something more I can do on the numerical side to, on reduce fill-in and/or help the iterative solver to converge faster? 
>
> I know my discretized PDE’s + boundary conditions are scaled consistently with regards to matrix entries.
> I’m using natural ordering (if my unknowns are a_ij, b_ij the unknown vector starts with a_00 b_00 a_10 b_10 a_20 b_20 and ends with a_nxny b_nxny…) but I do not think this has any impact?
>
> Thanks for your support,
>
>
> Thibaut