21st GAMM-Seminar Leipzig on
Robust Fast Solvers

Max-Planck-Institute for Mathematics in the Sciences
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  21st GAMM-Seminar
January, 26th-28th, 2005
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  Abstract Reinhard Nabben, Thu, 16.00-16.30 Previous Contents Next  
  Domain Decomposition Methods and Deflated Krylov Subspace Iterations
Reinhard Nabben (TU Berlin)

(joint work with C. Vuik, TU Delft)

The balancing Neumann-Neumann (BNN) and the BPS preconditioner are fast and successful preconditioners within domain decomposition methods for solving partial differential equations. For certain elliptic problems these preconditioners lead to condition numbers which are independent of the mesh sizes and are independent of jumps in the coefficients (BNN). Here we give an algebraic formulation of these preconditioner. This formulation allows a comparison with another solution or preconditioning technic - the deflation technic.

By giving a detailed introduction into the deflation technic we establish analogies between the the BNN-, the BPS-method and the deflation technic.

In the BNN- and the BPS-method special restriction and prolongation operators are used to solve coarse grid problems. Within the deflation operator these restrictions are build by so called deflation vectors to generate a subspace.

Using this analogies we can theoretically compare the BNN-, the BPS-method and the deflation method. 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 and the coarse grid correction preconditoner (BPS). Moreover, we establish a comparison of the A-norms of the iteration vectors generated by the preconditined CG-methods. We prove that deflation technic generates iteration vectors whose A-norms are less than the A-norms of the iteration vectors generated by the BNN-preconditioner.

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Last updated:
28.01.2005 Impressum
Concept, Design and Realisation
[O->]Jens Burmeister (Uni Kiel), Kai Helms (MPI Leipzig)
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