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of times the unknown is represented in the partitioning.
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The blockJacobi matrix with overlapping
blocks.
The blockJacobi iteration is often over or underrelaxed, using a relaxation parameter
' . The iteration can be de¬ned in the form
g
‚
˜ c
h p Y w'” v C C ” C '
t p D c r p 
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c sC
Recall that the residual at step is then related to that at step by
t e
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‚ c
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w…m1C”
C ”0
D c rpY hpY
C C 5w'”
4C
c sC
The solution of a sparse linear system is required at each projection step. These systems
can be solved by direct methods if the subblocks are small enough. Otherwise, iterative
methods may be used. The outer loop accelerator should then be a ¬‚exible variant, such as
FGMRES, which can accommodate variations in the preconditioners.
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In polynomial preconditioning the matrix is de¬ned by
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c }
D
v x
where is a polynomial, typically of low degree. Thus, the original system is replaced by
the preconditioned system
un j
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} }
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x x z
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}
which is then solved by a conjugate gradienttype technique. Note that and com x