On the multi-level splitting of finite element spaces
Numerische Mathematik
The construction of preconditioners for elliptic problems by substructuring. I
Mathematics of Computation
Iterative solution methods
On the abstract theory of additive and multiplicative Schwarz algorithms
Numerische Mathematik
Balancing domain decomposition for problems with large jumps in coefficients
Mathematics of Computation
Matrix computations (3rd ed.)
Journal of Computational Physics
Multigrid
Journal of Computational Physics
On the Construction of Deflation-Based Preconditioners
SIAM Journal on Scientific Computing
SIAM Journal on Matrix Analysis and Applications
Applied Numerical Mathematics - Developments and trends in iterative methods for large systems of equations—in memoriam Rüdiger Weiss
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
A Comparison of Deflation and Coarse Grid Correction Applied to Porous Media Flow
SIAM Journal on Numerical Analysis
On Generalizing the Algebraic Multigrid Framework
SIAM Journal on Numerical Analysis
A Comparison of Deflation and the Balancing Preconditioner
SIAM Journal on Scientific Computing
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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.