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As we all know, the Additive Schwarz approach can be used as either solver or preconditioner, however, my question is, what is the difference between the two? In other words, how to use AS as solver, how to use AS as preconditioner?

I found the below equations which give specific definitions of both, but I do not understand them, maybe some big guy can explain it a little bit?

  • Restricted Additive Schwarz (RAS)

Solver $$M_\text{RAS}^{-1}=\sum_{i=1}^NR_i^TD_i(R_iAR_i^T)^{-1}R_i$$ $$U^{n+1}=U^n+M_\text{RAS}^{-1}r^n,r^n:=F-AU^n$$ Precondition $$B^{-1}=M_\text{RAS}^{-1}$$

  • Additive Schwarz Method (ASM)

Solver $$M_\text{ASM}^{-1}=\sum_{i=1}^NR_i^T(R_iAR_i^T)^{-1}R_i$$ $$U^{n+1}=U^n+M_\text{ASM}^{-1}r^n,r^n:=F-AU^n$$ Precondition $$B^{-1}=M_\text{ASM}^{-1}$$

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By itself, Schwarz methods are stationary iterations just like Jacobi, Gauss-Seidel, or SOR. They converge to the solution, but often quite slowly.

But, like any other stationary method, one iteration (or a fixed, small number of iterations) can also used as a preconditioner in Krylov-space methods.

In other words, the distinction you are asking about is the same as between the Jacobi iteration as a solver, and the Jacobi preconditioner.

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  • $\begingroup$ Yes, you clearly described my doubts. Elsewhere, I found the following diagram to illustrate the correspondence between the stationary method as solver and preconditioner. You mentioned "In other words, the distinction you are asking about is the same as between the Jacobi iteration as a solver, and the Jacobi preconditioner.". To tell the truth, I am also trying to figure out the answer to this question, maybe you can explain a little, I just entered the line, thank you! In addition, you are very famous. I have been following deal.II for a long time. It is my honor for you to reply to me $\endgroup$
    – zhanghaoyuan
    Aug 27, 2021 at 7:21
  • $\begingroup$ @zhanghaoyuan In essence, stationary methods (i.e., fixed point iterations like Jacobi, SOR, SSOR, Gauss-Seidel) are not very good methods to solve linear systems since they converge quite slowly. That would be true for Schwarz methods as well. But they are reasonably good preconditioners because they are good approximations for the inverse matrix for at least part of the spectrum of the matrix. So they are widely used as preconditioners where they work far better than as solvers in their own right. You will find more information in many introductory numerical analysis text books. $\endgroup$
    – Wolfgang Bangerth
    Sep 8, 2021 at 15:29

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