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arXiv · 2607.15290

Matrix-Free FFT-HSS Preconditioning for Periodic Landau-Lifshitz-Gilbert Saddle-Point Systems

Hang Qi·Changqing Ye·Xiaofei Guan
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Abstract

In this paper, a matrix-free Hermitian/skew-Hermitian splitting (HSS) preconditioner is proposed for periodic Landau--Lifshitz--Gilbert (LLG) saddle-point systems. The main contributions are threefold. (1) The coupled skew/constraint block has an explicit \(4\times4\) nodal inverse and requires no local factorization. Combining this local inverse with FFT inversion of the shifted exchange block gives \(O(N_g\log N_g)\) work and \(O(N_g)\) temporary storage per application. (2) We establish well-posedness and an even-step GMRES residual bound, with an iteration estimate uniform in the mesh size and time step for a class of coupled refinements. (3) The projected implicit Euler and projection-free Crank--Nicolson-type midpoint discretizations generate saddle-point systems of the same form, so the same matrix-free FFT--HSS preconditioning procedure applies to both. They achieve first- and second-order temporal accuracy, respectively; for quadratic-affine energies, the midpoint scheme also preserves nodal length and satisfies an exact discrete dissipation identity. Two- and three-dimensional experiments confirm the predicted temporal orders and midpoint invariants. On the largest smooth tests, FFT--HSS reduces GMRES iterations by \(35\)--\(39\%\) relative to same-grid Householder preconditioning and yields approximately 15-fold and 3-fold speedups over unpreconditioned GMRES in two and three dimensions, respectively. Broadband tests retain mesh-independent iterations in the covered refinement regime at time steps 13.3 times the linearized explicit exchange limit.

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Hang Qi, Changqing Ye, Xiaofei Guan. 2026-06-24. Matrix-Free FFT-HSS Preconditioning for Periodic Landau-Lifshitz-Gilbert Saddle-Point Systems. https://arxiv.org/abs/2607.15290

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