Further comparison of additive and multiplicative coarse grid correction
A basic preconditioner is improved with a coarse grid correction and can be implemented either additively or multiplicatively like in the standard additive Schwarz method or in the balancing preconditioner, respectively. It is shown that the multiplicative variant has in fact always lower condition number when the basic preconditioner is appropriately scaled and assuming an appropriate scaling, 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. Two examples show 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.
scientific article; zbMATH DE number 6138081