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

Introduction

The interest in parallel solvers for partial differential equations has risen in recent years. Thus the domain decomposition method (DD) became increasingly popular since it contains a natural parallel structure. As it is usual with this method, we utilize the resulting block system of linear equations

$$\left( \begin{array}[c]{cc} { K_C } & { K_{CI} } \\ { K_... ...} {\underline{f}_C} \\ {\underline{f}_I} \end{array} \right)$$

(cf. section 3 for a more detailed description). The Additive Schwarz Method has been employed to construct preconditioners $C$ of the type

$$C := \left( \begin{array}[c]{cc} {I_C} & {K_{CI} B_I^{-T}}... ..._C} & {O} \\ {B_I^{-1}K_{IC}} & {I_I} \\ \end{array} \right)$$

for the (parallel) conjugate gradient method (cf. [12,13], and section 3). Two of the three components of this DD preconditioner, namely the (modified) Schur complement preconditioner $C_C$, and the local Dirichlet problem preconditioner $C_I$, have been studied intensively by the DD community (cf. the proceedings of the international Symposia on ``DD methods for partial differential equations'' since 1987 [7,3,4,8,,1,19], and also [2,5,6]). Haase/Langer/Meyer [12,13] proved that the quality of the DD preconditioner is dominantly influenced by the third component, the basis transformation $B_I$. This basis transformation determines the perturbation $T_C$ of the Schur complement which, in turn, influences the spectral radius $\, \mu = \varrho (S_C^{-1} T_C^{\phantom{\!\!\!\!-1}} ) \,$ and finally the relative condition number of the preconditioner, $\kappa(C^{-1}K) \, = \, O( \mu )$ (provided that $C_C$ and $C_I$ are chosen appropriately).

Several ideas to choose the basis transformation $B_I$ which is defined implicitly by some iteration method are summarized in section 3. Our interest is focused on the multigrid method for which it is known that

• the application of $\, s = O( \ln h^{-1} ) \,$ multigrid cycles ensures a bounded condition number $\, \kappa(C^{-1}K) \, = \, O( 1 ) \,$ (cf. [12]).

• the application of $\, s = 1 \,$ multigrid cycle results (in numerical experiments) in a growing condition number $\, \kappa(C^{-1}K) \, = \, O( h^{-1} )$ as $h \to 0 \,$ [12].

In this paper we investigate the use of one full multigrid cycle for the definition of the basis transformation. Special interest is paid firstly as to whether the increasing condition number $\, \kappa(C^{-1}K) \,$ can be overcome as $h \to 0$, and secondly to the computational expense.

Sections 2 is devoted to the model problems whereas section 3 deals with the DD preconditioner and the basis transformation $B_I$. The theoretical analysis of the full multigrid basis transformation is performed in section 4. Unfortunately the analysis turned out to be so technical that only the full two-grid operator on a simple model problem could be treated. In section 5 the numerical experiments with the full multigrid operator are given. Section 6 summarizes the results obtained.

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