FGMRES
Flexible Generalized Minimal Residual method (FGMRES) is an iterative Krylov subspace method for solving linear systems A x = b, especially when A is non-symmetric and preconditioning is beneficial but may vary during the solve. It extends GMRES by allowing a nonstationary preconditioner, meaning the preconditioner can change from one iteration to the next.
In standard GMRES, a fixed preconditioner M is applied uniformly. In FGMRES, a sequence of preconditioners M1,
Because the preconditioner can vary, convergence behavior can be more favorable for difficult problems, but implementation
FG MRES is well suited for large, sparse, non-symmetric systems where preconditioning quality changes over time