# What is the difference between Adittive Schwarz as a preprocessor and a solver?

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 $$B^{-1}=M_\text{ASM}^{-1}$$