5.7.1 Sequential algorithm

For notation purposes of the multigrid algorithm we have to introduce the following vectors and matrices :

$I_{q}^{q-1}$

Restriction/Projection operator for mapping fine grid data onto the coarse grid.

$I_{q-1}^{q}$

Interpolation operator for mapping coarse grid data onto the fine grid.

$S_{_{\mathrm{pre}}}^{\nu_{\mathrm{pre}}}$

Presmoothing operator reducing  $e_{\mathrm{high}}$, applied $\nu_{\mathrm{pre}}$-times.

$S_{_{\mathrm{post}}}^{\nu_{\mathrm{post}}}$

Postsmoothing operator reducing  $e_{\mathrm{high}}$, applied $\nu_{\mathrm{post}}$-times.

$\underline{d}_{q}$

Defect on $q^{\text{th}}$ grid.

$\underline{w}_{q}$

Correction on $q^{\text{th}}$ grid.

   MGM$^\gamma$

Recursive multigrid procedure, applied $\gamma$-times
($\gamma=1$ - V-cycle, $\gamma=2$ - W-cycle)

We can choose the iteration methods from Sec. 5.3 - 5.5 as presmoother or postsmoother, respectively.

If one uses multigrid as preconditioner in a CG algorithm then the proper multigrid operator has to be symmetric. In that case we have to choose the grid transfer operators, i.e., interpolation and restriction, and the smoothing such that $I_{q}^{q-1} = \left(I_{q-1}^{q}\right)^T$ and $S_{_{\mathrm{pre}}}= \left(S_{_{\mathrm{post}}}\right)^T$ with $\nu_{\mathrm{pre}} = \nu_{\mathrm{post}}$ holds. The defect system on the coarsest grid will be solved directly.