[petsc-users] Correlation between da_refine and pg_mg_levels

[Barry Smith]

> On Mar 31, 2017, at 10:00 AM, Jed Brown <jed at jedbrown.org> wrote:
> 
> Justin Chang <jychang48 at gmail.com> writes:
> 
>> Yeah based on my experiments it seems setting pc_mg_levels to $DAREFINE + 1
>> has decent performance.
>> 
>> 1) is there ever a case where you'd want $MGLEVELS <= $DAREFINE? In some of
>> the PETSc tutorial slides (e.g., http://www.mcs.anl.gov/
>> petsc/documentation/tutorials/TutorialCEMRACS2016.pdf on slide 203/227)
>> they say to use $MGLEVELS = 4 and $DAREFINE = 5, but when I ran this, it
>> was almost twice as slow as if $MGLEVELS >= $DAREFINE
> 
> Smaller coarse grids are generally more scalable -- when the problem
> data is distributed, multigrid is a good solution algorithm.  But if
> multigrid stops being effective because it is not preserving sufficient
> coarse grid accuracy (e.g., for transport-dominated problems in
> complicated domains) then you might want to stop early and use a more
> robust method (like direct solves).

Basically for symmetric positive definite operators you can make the coarse problem as small as you like (even 1 point) in theory. For indefinite and non-symmetric problems the theory says the "coarse grid must be sufficiently fine" (loosely speaking the coarse grid has to resolve the eigenmodes for the eigenvalues to the left of the x = 0).

https://www.jstor.org/stable/2158375?seq=1#page_scan_tab_contents