Fast adaptive algorithms for micromagnetics

Evaluation of the long-range magnetostatic field is the most time-consuming part in a micromagnetic simulation. In a magnetic system with N particles, the traditional direct pairwise summation method yields O(N²) asymptotic computation time. An adaptive fast algorithm fully implementing the multipole and local expansions of the field integral is shown to yield O(N) computation time. Fast Fourier transform techniques are generalized to entail finite size magnetic systems with nonperiodic boundary conditions, yielding O(N log₂ N) computation time. Examples are given for calculating domain wall structures in Permalloy thin films. The efficiency of the fast Fourier transform makes it almost always the faster method for any large-size system, while the multipole algorithm remains effective for more complex geometries and systems with highly irregular or nonuniform particle distributions