[petsc-users] How to implement projection preconditioner?

[Jed Brown]

On Fri, Jun 15, 2012 at 2:23 PM, Abdul Hanan Sheikh
<hanangul12 at yahoo.co.uk>wrote:

> I recall my previous thread; copy and paste here, and then I try to ask:
>
> Suppose C = P*A_H^{-1}*R is the coarse grid solver. Then with a
> post-smoother S, application of the preconditioner to the vector "b" would
> be C b + S (b - A C b). If we drop the post-smoother by setting S = I, We
> get Prec = C b + b - A C b. Note that there is no presmoother.
>
> This is the preconditioner I have used before, in PCMG framework, with no
> pre-smoothing and Richardson(without any preconditioner) as post smoother.
>
> I need to apply the updated version of above Prec = C b + (b - A C b),
> where A is replaced by M^-1 A , and A_H by M_H^-1 A_H .
>

With left preconditioning, you also have to apply the preconditioner to the
right hand side, thus making it look just like a smoother.


>
> Now does it make sense :
>
> 1)  incorporating M as only preconditioner in presmoother will replace A
> by M^-1 A in Prec = C b + (b - A C b) ?
>

No, it uses the preconditioner the way that preconditioners work, see above.


> 2) incorporating M as preconditioner to CG-solver will replace A_H by
> (M_H^-1 A_H ) ? I am kinda of sure about this point 2.
>

C already contains A_H^{-1}. Replacing that by M_H^{-1} is an approximate
coarse level solve (that's okay and seems to be what you want, just not
what you have written, which is not a consistent method).
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