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.



1 Answer 1


Set $e=0$.

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


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