[petsc-users] boomerAmg scalability

[Jed Brown]

On Mon, Dec 19, 2011 at 18:48, Mohamad M. Nasr-Azadani <mmnasr at gmail.com>wrote:

> I have been using BoomerAMG as a preconditioner joint with an iterative
> solver, e.g. GMRES of BiCGs for regular 3D CFD problems.
> On the top of my head, I can not remember if I had the strong scaling
> tests done (I will look into it and let you know if you found any), but for
> the weak-scaling case, I definitely saw some scaling issues.
> As the size of the system increases, the number of iterations does also
> increase (somewhat significantly for my test problem, i.e. incompressible
> N-S with complex geometry) which ultimately deteriorates the weak-scaling
> behaviors.
>

It is very dangerous to use AMG directly on the indefinite problem produced
by most discretizations of incompressible flow. For example, for mixed
finite element methods, I have on multiple occasions observed BoomerAMG
produce a singular preconditioner (with huge null space), leading to the
appearance of convergence in the preconditioned norm, but no actual
convergence.


>
> This is also reported in the reports given by the hypre team, cf.
> https://computation.llnl.gov/casc/linear_solvers/pubs/pmis_report.pdf
>
> (cf. see Table 6.1 for the Stokes flow simulation results and scaling).
>

This table is for a very special discretization, First Order System Least
Squares (FOSLS). Advocates of FOSLS like to point out that the method does
not require inf-sup compatibility between velocity and pressure spaces, so
equal-order spaces can, in principle, be used, without harming the
"optimal" convergence rates. The problem is that equal order spaces, which
are almost always used in practice, cause systematic conservation errors.
Even for very simple geometries, "incompressible" flow solutions can
exhibit more than 90% mass loss. If you look at the literature, you will
notice that many of the most prominent advocates of FOSLS were not
forthcoming about this "detail" for the first decade of their publications.
In the last five years, they have published on "enhanced mass conservation"
techniques, with which, still for simple, well-resolved flow problems, they
manage to get less than 20% mass loss with linear elements and less than 1%
by using higher order elements (up to quartic). I have yet to meet an
engineer who would consider such systematic mass loss acceptable.
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