Assume I have 3 levels of grids. Finest Grid = level 2, Coarser Grid = level 1, Coarsest Grid = level 0.
- Relax $u$ on $Au = b$ at level 2 for 3 times.
- Find residual $r2$ at level 2, then restrict to level 1
- Relax $e$ on $Ae=r$ at level 1 for 3 times.
- Find residual $r1$ at level 1, then restrict to level 0.
- Solve $Ae=r$ on level 0 till convergence.
- Interpolate $e$ on level 0 to level 1.
- Correct $e$ on level 1 and relax $e$ 3 more times.
- Interpolate $e$ on level 1 to level 2.
- Correct $u$ on level 2 and relax $u$ 3 more times.
- Check if current error is $>$ tolerance (assume yes)
- 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.