Page 143 - Computational Fluid Dynamics for Engineers

4.5 Finite-Difference Methods for Elliptic Equations 129

and has the important property

where c is constant. The above interpolation and restriction operators can be
extended to general coarse and fine grids, including unstructured grids.
The basic idea of the Multigrid method for
A hV h = ¥h (4.5.42)

has the following steps.
1. Relax n\ times on A^U^ = F^ on Q^ with initial guess U h
2. Compute
h
r2h = j2h( Fh _ A hV ) (4.5.43a)
Solve
&2hU 2h = v 2h on Q 2h (4.5.43b)
3. Correct fine grid approximation:

jjh = jjh + jh^h (4.5.44)
4. Relax ri2 times on A^U^ = F^ on Q^ with initial guess U^

The step 2 for coarse grid can be carried out in several ways. Three popular
procedures, called V-cycle, W-cycle and Fnll Multigrid (FMV), are shown in
Figs. 4.11, 4.12 and 4.13. In the V-cycle, the calculations begin at A with the
solutions obtained in step 1 carried out on a fine grid Q^. In B the calculations
are performed on the coarse grid i?2/i to determine the residual r 2h so that u 2h
on Q 2h c a n be determined from Eq. (4.5.43b). This procedure in B is repeated
for the same number iterations in step 1 at A. In C, the procedure in B is
repeated on grid i?4^. In D, on the coarsest grid, Eq. (4.5.43b) is solved with a
direct method.
In E, F and G, the corrections are made to the solutions obtained in D,
according to Eq. (4.5.44), for example, in E, the correction is

u 4/l = u 4h + i££u 8h

Ih

Ah

D 8/i Fig. 4.11. V-cycle on four level grids.