# Increasing V-cycles for constant Coarsest Grid Size and increasing Fine Grid size

Problem statement

I implemented geometric multigrid for $-\nabla^{2}=f$ where $f=\frac{3\pi^{2}}{4}sin \frac{\pi x}{2} sin \frac{\pi y}{2} sin \frac{\pi z}{2}$ on $\Omega \in [0,1]$ on a unit cube. Dirichlet boundaries on the left face, bottom face and front face is 0. Neumann boundaries on the top, right and back face are $\frac{\partial u }{\partial n} = 0$.

Method

A multigrid method is used to solve the equation. I approximate the ghost points at the Neumann boundary using the central difference formula.

Method overview (from comments, confirmed by the author): Start from the fine mesh (the final mesh the equation to be solved with), proceed to the coarser mesh to compute the correction, propagate it back and smooth at the end of the multigrid procedure.

Observations

The problem is when I fix my coarsest grid (say to 16x16x16) and measure V-cycles for increasing fine grid sizes, my V-cycles are not constant. I read in the book MULTIGRID by Trottenberg et. al. that we need to use a modified Full Weighted restriction operator to prevent incorrect scaling at Neumann boundaries. Further, I am unable to understand this modified full restriction operator mentioned in the book.

In another example where I implemented a mixed Dirichlet-Neumann problem $-\nabla^{2}=0$ where $u = 1 + x + y + z$ at Dirichlet boundaries I did not need to use this modified operator for convergence (fixed coarsest grid and increasing finest grid, the V-cycles remained constant).

Question

Could the "modified Full Weighted restriction" be causing a deterioration in the convergence rate?