# A doubt in Multigrid V-cycle

Assume I have 3 levels of grids. Finest Grid = level 2, Coarser Grid = level 1, Coarsest Grid = level 0.

1. Relax $u$ on $Au = b$ at level 2 for 3 times.
2. Find residual $r2$ at level 2, then restrict to level 1
3. Relax $e$ on $Ae=r$ at level 1 for 3 times.
4. Find residual $r1$ at level 1, then restrict to level 0.
5. Solve $Ae=r$ on level 0 till convergence.
6. Interpolate $e$ on level 0 to level 1.
7. Correct $e$ on level 1 and relax $e$ 3 more times.
8. Interpolate $e$ on level 1 to level 2.
9. Correct $u$ on level 2 and relax $u$ 3 more times.
10. Check if current error is $>$ tolerance (assume yes)
11. Relax $u$ again 3 times on level 2, calculate residual, then restrict.

Clearly at this stage at level 1 matrix $r$ i.e. $Ae=r$ has changed. So before we relax on level 1, do we set $e=0$ ? Or do we continue with the previous value of $e$ ? I think $e$ should be reset to 0 as the previous calculations were according to a different $r$ (restricted residual) matrix.

Thanks.

Set $e=0$.
The previous $e$ there has already been "accounted for," as you have used it to correct your solution.