[firedrake] High-contrast Stokes problem

[Loic Gouarin]

Thanks Lawrence for your answer.

I works perfectly and we can't do more simple !!!

I have a last question. I don't want to use a Q2-Q1 elements for the 
Stokes problem but a 4Q1-Q1 elements which means that linear elements 
for velocity live on a mesh that is once more refined globally than the 
elements for pressure.

When I do that:

N = 64
meshu = UnitSquareMesh(2*N, 2*N, quadrilateral=True)
meshp = UnitSquareMesh(N, N, quadrilateral=True)
V = VectorFunctionSpace(meshu, "DQ", 1)
W = FunctionSpace(meshp, "DQ", 1)
Z = V * W

I have an error:
ValueError: All function spaces must be defined on the same mesh!

Thanks,
Loic

Le 06/02/2017 à 11:59, Lawrence Mitchell a écrit :
> Hi Loic,
>> On 6 Feb 2017, at 10:31, Loic Gouarin <loic.gouarin at math.u-psud.fr> wrote:
>>
>> Hi,
>>
>> I would like to solve a Stokes problem with a viscosity which is not constant and could be high in some parts of the domain (as in sinker problem). The solvers that I want to use are in this article:
>>
>> http://dx.doi.org.ezproxy.math.cnrs.fr/10.1016/j.cam.2013.10.016
>>
>> I would like to use a P0 interpolation for the viscosity function.
>>
>> Could you tell me how can I set this function into my variational formulation ?
>
> You can make a P0 space into which you interpolate your viscosity, and then use that in the form:
>
> P0 = FunctionSpace(mesh, "DG", 0)
>
> nu = Function(P0)
>
> nu.interpolate(whatever)
>
> Now use nu as normal:
>
> a = nu*inner(grad(u), grad(v))*dx ...
>
>
>> The other question is to construct the S matrix when you use the Schur complement as a preconditioner. I have to set the pressure mass matrix for S and scale it with the inverse of the viscosity. How can I do that ?
> You can provide a separate form that is used to construct the preconditioning operator.  Say something like:
>
> aP = nu*inner(grad(u), grad(v))*dx + (1/nu)*p*q*dx
>
> solve(a == L, w, Jp=aP)
>
> And pass solver parameters:
>
> -pc_type fieldsplit
> -pc_fieldsplit_type schur
> # We didn't provide off diagonal blocks in the preconditioning
> # matrix
> -pc_use_amat
> # Use a11 block from aP to provide preconditioning operator for S.
> -pc_fieldsplit_schur_precondition a11
>
> If you have really high contrast (and/or many sinkers), you will probably find that the viscosity-weighted pressure mass matrix gives mesh independent convergence, but quite high iteration counts.  In those case I think you want one of the "BFBt"-like preconditioners (e.g. https://arxiv.org/abs/1607.03936)
>
> Cheers,
>
> Lawrence
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http://www.math.u-psud.fr/~gouarin
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