I have a nicely working multigrid solver, which I use for solving the Poisson equation from an electrostatic problem. I solve this equation first without any charges, and then many times with a slowly changing charge distribution.

Right now, I do a FMG from scratch each time I solve this equation, ignoring the fact that I already have a previous solution which could be updated instead of starting new from scratch. But if I use a V- or a W-cycle for this, I fear that there might be situations (if the charge distribution changed significantly) where this is worse than just starting from scratch with FMG.

I could try to apply FMG to the difference between my old solution and the final solution. However, this might be wasteful (in terms of memory), because I would have to store both the old solution, the real right hand side, the right hand side for the difference, and the difference on the finest grid. Hence I wonder whether using the F-cycle instead wouldn't be just as good, or even equivalent to this procedure.

  • $\begingroup$ I now use a W-cycle. It works quite well by itself, so no need to worry about whether it is better or worse than FMG. Maybe an F-cycle would be fine as well, but W-cycles are quite standard, and normally better than V-cycles. $\endgroup$ Jan 17, 2016 at 22:09

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I would like to approach this problem from the perspective of the simulation of an incompressible flow. Pressure is governed by a Poisson's equation and as the simulation progresses the source term of this equation changes as it is dependent on the evolving velocity field. In my experience I found that retaining the old guess for the pressure does not slow down the solution process or change the solution. Could you test whether retaining the previous solution slows the convergence on the next round of FMG cycle where the charge distribution is different? I infer from your question that this test has not been performed.

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    $\begingroup$ How can I retain the old guess with FMG? Does it mean to first compute the residuum of the old guess on the finest grid, and then do a very accurate restriction of the residuum to the coarsest grids without any intermediate smoothing? But even then, I'm in a different situation than when I start from scratch, because all boundary conditions on the coarser grid are now zero, while they may be non-zero if I start from scratch. $\endgroup$ Apr 22, 2015 at 11:18
  • $\begingroup$ I am sorry, I misread the situation. In FMG method using V or W cycle one starts with the exact solution on the coarsest grid. This is solved using a direct LU based method and the V or W cycles are run progressively reaching the final V-cycle where the equation is solved on the finest grid. Retaining the solution from the previous iteration does not help here. And there is no way one can do it as it gets overwritten by the exact solution computed on the coarsest grid before entering the FMG cycle using V or W cycle. I think this situation does not change even if you switch to an F-cycle. $\endgroup$
    – Trinath
    Apr 23, 2015 at 5:14

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