poisson problem with periodic boundary conditions

[Ryan Abernathey]

Hello,

I am new to pyamg. I would like to solve a poisson problem with periodic boundary conditions (i.e. the domain wraps around in the x and y directions). I have tried building up the correct A matrix myself, but I am having trouble getting the solver to converge.

Can anyone advise me on the best way to approach this problem?

Thank you very much!

-Ryan Abernathey

[Jacob Schroder]

Hi Ryan,

If your Poisson problem is periodic in all directions, then the
constant vector will likely be in the nullspace of the matrix, i.e.,
you have a singular matrix, A. This would explain very slow
convergence. Essentially multigrid can't reduce error, if that error
e satisfies A*e = 0. To check this, see if
>>> A*numpy.ones((A.shape[0],0))
gives you a vector of essentially zero values. What to do, is to use
the callback parameter offered by pyamg.krylov.cg() or
smoothed_aggregation_solver.solve(). This parameter lets you apply a
user-defined function to the current solution guess every iteration.
Set callback to be a function that will project out the constant
vector every few iterations. Convergence should happen quickly.

Hope this helps,

Jacob

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