[petsc-users] Stokes-Brinkmann equation preconditioner

[Matthew Knepley]

On Fri, Oct 4, 2019 at 6:04 AM Lawrence Mitchell <wence at gmx.li> wrote:

>
>
> > On 4 Oct 2019, at 10:46, Matthew Knepley via petsc-users <
> petsc-users at mcs.anl.gov> wrote:
> >
> > On Thu, Oct 3, 2019 at 6:34 PM Salazar De Troya, Miguel via petsc-users <
> petsc-users at mcs.anl.gov> wrote:
> > I am trying to solve the Stokes equation with the Brinkman term to
> simulate a solid material. My intention is to implement the preconditioner
> in this paper:https://onlinelibrary.wiley.com/doi/epdf/10.1002/fld.426
> (section 2.6)
> >
> >
> > Link does not work for me.
>
> Try https://onlinelibrary.wiley.com/doi/epdf/10.1002/fld.426 (mail
> clients are awful)
>
> >
> > where they solve for the velocity and substitute that expression in the
> pressure equation. They end up solving a system of the type B K^-1 B^T,
> i.e. the Schur complement of the problem. For this system of equations,
> they argue that the preconditioner in page 11 is perfect for a given
> constant Brinkman penalty term.
> >
> >
> >
> > Because I am solving for velocity and pressure without doing any
> substitution, I thought I could use a PC fieldsplit type Schur (full
> factorization)
> >
> >
> > Yes, this will form the exact factorization and a matrix-free form of
> the Schur complement.
> >
> > and provide the preconditioner in the paper to solve the Schur
> complement.
> >
> >
> > Yes, you can provide a user-defined PC for the Schur complement.
> >
> > My question is, should I provide this preconditioner through
> PCFieldSplitSetSchurPre or through fieldsplit_1_pc_type (probably through
> the Firedrake interface as inhttps://
> www.firedrakeproject.org/demos/stokes.py.html) ?
> >
> >
> > The name PCFieldSplitSetSchurPre seems to be very misleading. You do not
> use it to provide a _preconditioner_. You use it to determine
> > the _preconditioning matrix_ from which the actual preconditioner is
> built. The preconditioner itself is defined using -fieldpslit_1_pc_type.
> > Since I do not know what the preconditioner looks like, I cannot say
> what preconditioner matrix you would want. Since Firedrake can construct
> > any operator for you, you might not care about the matrix we pass to you.
>
> The action of the PC is defined by:
>
> M^{-1} q = \phi_q + \mu q
>
> \mu is some number, and \phi_q solves the BVP
>
> -div \alpha^{-1} grad \phi_q = q
>
> grad \phi_q \dot n = 0
> \int \phi_q = 0
>
> So your PC for the fieldsplit_1 needs to:
>
> 1. Discretise and solve this BVP to compute \phi_q
> 2. Add \mu q.
>
> I.e. the action of PCApply is:
>
> y <- \mu x + Laplace^{-1} x
>
> (Aside, surely a primal-dual error has been made in the analysis here)
>
> You could, I think, do this by providing the discretisation of this
> laplacian and using an additive PCComposite (although I don't know what PC
> you would use to just scale the input, it's easy to write one though).
>

Richardson R(x, y) is

   y += mu (b - Ax)

so if b = 0 and A = I, you get that. It seems easier just to make a PCSHELL
with VecAXPY I guess.

  Thanks,

      Matt


> Cheers,
>
> Lawrence
>
>
>

-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener

https://www.cse.buffalo.edu/~knepley/ <http://www.cse.buffalo.edu/~knepley/>
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