A comparison of deflation and the balancing Neumann-Neumann preconditioner

A comparison of deflation and the balancing Neumann-Neumann preconditioner

Nabben, R.
Vuik, C.

Electrical Engineering, Mathematics and Computer Science

2004

In this paper we compare various preconditioners for the numerical solution of partial differential equations. We compare the well-known balancing Neumann Neumann preconditioner used in domain decomposition methods with a so-called deflation preconditioner. We prove that the effective condition number of the deflated preconditioned system is always, i.e. for all deflation vectors and all restrictions and prolongations, below the condition number of the system preconditioned by the balancing Neumann-Neumann preconditioner. Even more, we establish that both preconditioners lead to almost the same spectra. The zero eigenvalues of the deflation preconditioned system are replaced by eigenvalues which are one if the balancing Neumann-Neumann preconditioner is used. Moreover, we proved that the A-norm of the errors of the iterates build by the deflation preconditioner is always below the A-norm of the errors of the iterates build by the balancing Neumann-Neumann preconditioner. Additionally, the amount of work of one iteration of the de ation preconditioned system is less than the amount of work of one iteration of the balancing Neumann-Neumann preconditioned system. Finally, we establish that the deflation preconditioner and the balancing Neumann-Neumann preconditioner produces the same iterates if one uses certain starting vectors. Numerical results for porous media flows emphasize the theoretical results.

deflation
coarse grid correction
balancing Neumann Neumann
preconditioners
Conjugate Gradients
porous media ow
scalable parallel preconditioner

http://resolver.tudelft.nl/uuid:f0ad06a2-19e4-4344-adb9-24d69cdd57f6

Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics

1389-6520

Reports of the Department of Applied Mathematics, 04-09

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(c) 2004 R. Nabben; C. Vuik