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, M2, ..., Mk can be used, with corresponding preconditioned vectors z_j = M_j^{-1} v_j based on the current direction v_j. The method builds a flexible Arnoldi process where A Z_j = Z_{j+1} H_j, with Z_j containing the preconditioned vectors z_j and H_j an upper Hessenberg matrix. The iterate x_j is formed as x_j = Z_j y_j, where y_j minimizes the residual over the generated subspace. Practically, a small least-squares problem is solved at each step to update the residual norm.

Because the preconditioner can vary, convergence behavior can be more favorable for difficult problems, but implementation details become more involved than standard GMRES. As with GMRES, FGMRES commonly uses restarts to control memory usage, denoted FGMRES(m) for a restart after m iterations. Each iteration may include inner solves or different preconditioning techniques, which can reflect evolving physics, time dependence, or domain-decomposition strategies.

FG MRES is well suited for large, sparse, non-symmetric systems where preconditioning quality changes over time or across iterations. It shares the general framework of GMRES and Krylov methods, while offering flexibility to accommodate nonuniform preconditioning strategies. See also GMRES, Arnoldi process, and preconditioning.