Hi Loic,
On 6 Feb 2017, at 10:31, Loic Gouarin <loic.gouarin@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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