Further comparison of additive and multiplicative coarse grid correction

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• 作者：Yvan Notay ; ¹ ; ^{ynotay@ulb.ac.be} ; Artem Napov² ; ^{anapov@ulb.ac.be}
• 关键词：Multigrid ; Domain decomposition ; Convergence analysis ; Coarse grid correction ; Preconditioning
• 刊名：Applied Numerical Mathematics
• 出版年：2013
• 出版时间：March, 2013
• 年：2013
• 卷：65
• 期：Complete
• 页码：53-62
• 全文大小：205 K

文摘

We consider the situation where a basic preconditioner is improved with a coarse grid correction. The latter can be implemented either additively (like in the standard additive Schwarz method) or multiplicatively (like in the balancing preconditioner). In a previous study, Nabben and Vuik compare both variants, and state that a theoretical comparison of the condition numbers is not possible: whereas it is admitted that the condition number is in most cases smaller with the multiplicative variant, they provide an example for which the converse is true. Here we show that the multiplicative variant has in fact always lower condition number when the basic preconditioner is appropriately scaled. On the other hand, we also show, again assuming an appropriate scaling, that the condition number of the additive variant is at worst a modest multiple of that of the multiplicative variant. Hence both approaches are qualitatively equivalent. Eventually, we show with some examples that both the upper and lower bounds on the condition number of the additive variant are sharp: it can be in some cases equal to the condition number of the multiplicative variant, and in other cases arbitrarily close to the aforementioned modest multiple of this latter value.