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
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
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
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