Geometric multigrid - FAS

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Geometric Multigrid or (FAS)

In Geometric multigrid a hierarchy of meshes is generated.The discretized equations are evaluated on every level. The advantage of geometric multigrid over algebraic multigrid is that the former should perform better for non-linear problems since non-linearities in the system are carried down to the coarse levels through the re-discretization.

Restriction and Prolongation Operators

Geometric multigrid requires restriction of both the fine grid solution and its residual or defect to coarse level. The restriction operator that transfers the solution to the next coarser grid level can be formed using a full-approximation scheme. In this, the solution for a coarse cell is obtained by taking the volume average of the solution values in the embedded fine grid cells. Or by other means of weighting. Residuals for the coarse grid cell are obtained simply by summing the residuals in the embedded fine grid cells.

Two grid Cycle

The idea of geometric multigrid could be made clear by this two grid algorithm.

$a_P \phi _P = \sum\limits_{nb} {a_{nb} \phi _{nb} + s}$

Then at fine level 2

$a_P^2 \phi _P^2 = \sum\limits_{nb} {a_{nb}^2 \phi _{nb}^2 + s^2 } + r^2$

this gives at coarse level 1

$a_P^1 \phi _P^1 = \sum\limits_{nb} {a_{nb}^1 \phi _{nb}^1 + s^1 } + \left[ {\bar a_P^1 \bar \phi _P^1 - \sum\limits_{nb} {\bar a_{nb}^1 \bar \phi _{nb}^1 - \bar s^1 - \bar r^1 } } \right]$

Where overbar represents restricted variables. The additional source $\left[ {\bar a_P^1 \bar \phi _P^1 - \sum\limits_{nb} {\bar a_{nb}^1 \bar \phi _{nb}^1 - \bar s^1 - \bar r^1 } } \right]$ does not change during the relaxations at coarse level.

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