I have a question regarding the FMG (Full Multi Grid) performance while computing Poisson's equation using Higher Order Compact discretization. I am using a sixth order compact scheme to discretize the Poisson's equation and use a Gauss Seidel method for smoothing/relaxing. And I employ Neumann boundary conditions on all sides of the 2D domain I am dealing with.

I followed Multigrid tutorial by Briggs et al to deal with the Neumann BCs by making the null-space vector orthogonal to the RHS vector in the equation Ax = b. I do not pin a value in the domain as it would lead to complications in transfers between multigrid levels. Instead I impose the zero mean condition on every level of the FMG cycle.

For a 128X128 grid the L2 norm of the residual stands at 32.00 at the end of FMG cycle which is very high for the entire domain. But it is less when distributed evenly among all the interior points. I would like converge the L2 norm atleast to 1e-2 for the Neumann case. Are there any techniques that I can use to speed this up to the said convergence without changing the number of times I relax on the fine grid. The higher order scheme itself has bad smoothing properties I think. It is not as effective in culling the high frequencies present in the domain.

Is there an effective and clear strategy to pin the solution at one point int he fine grid and continue with FMG?

Also, will it help if we use a higher order interpolation operator from coarse to fine grids when we go from one V-cycle to the other?

Please let me know your suggestions.

Best Trinath