GMRES convergence in highly anharmonic low-symmetry phases

Determine whether highly anharmonic low-symmetry phases require more GMRES iterations to converge in the iterative solution of the SSCHA free-energy-Hessian linear system, owing to the need to relax polarization vectors relative to the auxiliary SSCHA dynamical matrix.

Background

The paper reformulates the SSCHA free-energy-Hessian calculation as a linear system and solves it with the GMRES algorithm. Because the linear operator is not generally positive definite near structural instabilities, conjugate-gradient methods are not applicable in general; a harmonic preconditioner is therefore used to accelerate convergence.

The authors report empirically that 10–30 GMRES steps typically suffice and that this count is largely independent of mesh size. However, they explicitly leave unresolved whether highly anharmonic low-symmetry phases may require more iterations, potentially because their polarization vectors must relax relative to those of the auxiliary SSCHA dynamical matrix. This is a concrete unresolved question about the robustness and scalability of the solver.