Multigrid Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 230

Fig. 49. A coarse triangulation for which one of the (coarse) triangles has been
decomposed into 16 subtriangles in two steps. The other triangles should be
decomposed in the analogous way.

We now define the multigrid method for conforming elements recursively .

1.5 Multigrid Iteration MGM ( k -th cycle at level ≥ 1):

Let u ,k be a given approximation in S .
1. Pre-Smoothing. Carry out ν 1 smoothing steps:

u ,k, 1

ν 1 u ,k .

= S ( 1 . 6 )

v − 1 denote the solution of the variational prob-

2. Coarse-Grid Correction. Let

lem at level − 1,

J(u ,k, 1

+ v) −→ min

v ∈ S − 1

! ( 1 . 7 )

= v − 1 .
If > 1, compute an approximation v − 1

1, find the solution and set v − 1 If =

v − 1

of by carrying out µ steps of

MGM − 1 with the starting value u − 1 , 0

= 0.

Set

u ,k, 2 = u ,k, 1 + v − 1 .

( 1 . 8 )

3. Post-Smoothing. Carry out ν 2 smoothing steps,

u ,k, 3

ν 2 u ,k, 2 ,

= S

and set u ,k + 1 = u ,k, 3 .

1.6 Remarks. (1) If only two levels are being used, then we have only the case
=

1, and the coarse-grid correction will be done exactly. For more than two
levels, we compute the solution on the coarse grid only approximately, and for
the convergence theory we treat the multigrid iteration as a perturbed two-grid
iteration.