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

  
Summary

Aim of this work was the analysis of efficient, cheap and parallel preconditioners $C$ based on domain decomposition and especially on the Additive Schwarz Method. Starting with known theoretical and practical results the importance of the occurring basis transformation $B_I$ has been pointed out in section 3. The influence of $B_I$ on $\mu =\varrho \, (S_C^{-1} T_C^{\phantom{\!\!\!\!-1}})$ and thus on $\kappa(C^{-1} K)$ and the number of cg iterations has been shown. Our investigation has been focused on the full multigrid method in order to define $B_I$, with special emphasis on the behaviour of $\mu =\varrho \, (S_C^{-1} T_C^{\phantom{\!\!\!\!-1}})$ as $h \to 0$.

The model problem analysis of section 4 could only be carried out for the full two-grid operator and the model problem 1. The numerical analysis suggested strongly $\mu = O( h^{-1} )$. A proof has been established for a smoothing parameter $\omega = 1.0$.

The first part of the numerical experiments verified our analysis. In the second part the full multigrid and the multigrid basis transformation were compared for both model problems. As anticipated the cg iteration numbers grew slower with the first method. Additionally the strong dependence of the iteration numbers on the triangulation and the smoothing parameter $\omega$ has been pointed out.

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