HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-980724-A

  
Comparison of multigrid versus full multigrid

These practical computations concentrate on the number of pccg iterations. The three components $B_I, C_I$ and $C_C$ of the preconditioner (1) have been discussed briefly in section 3. Here they are defined as follows:

• $B_I$ is defined by multigrid or full multigrid, respectively. We employ the V cycle with one $\omega$ Jacobi pre- and post-smoothing sweep.

• $C_I$ is defined by the W multigrid cycle with two Gauss Seidel pre- and post-smoothing sweeps.

• For $C_C$ the Dryja preconditioner [5] and the BPS preconditioner (cf. [2]) are utilized in model problem 1 and 2, respectively.

Note that (in contrast to the model problem analysis) the multigrid algorithms use the usual FEM interpolation and restriction between successive grids.

In our tests we compare the multigrid and the full multigrid method for defining the basis transformation $B_I$. Stimulated by the results of the full two-grid operator different parameters of the $\omega$-Jacobi smoother are investigated.

Additionally we apply exact solvers $B_I = C_I := K_I$. Then $C_C$ solely influences the preconditioner $C$, and the iteration numbers obtained here thus measure the spectral equivalence constants of $C_C$ to $S_C$.

The MG and FMG methods are performed on 2 to 8 nested grids. In our numerical tests two different types of triangulation are considered for either model problem. All four coarsest grids are depicted in Figures 4 and 5. The standard triangulation basically confirms the theoretical analysis but the criss-cross pattern illustrates the strong influence of the triangulation on the iteration numbers and the optimal smoothing parameter $\omega$.

  

Figure 4: Coarse grids of model problem 1, standard and criss-cross triangulation

  

Figure 5: Coarse grids of model problem 2, standard and criss-cross triangulation

HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-980724-A