Computational micromagnetics in three dimensions is of increasing interest with the development of magnetostrictivesensors and actuators.

In solving the Landau-Lifshitz-Gilbert (LLG) equation, the governing equation of magneticdynamics for ferromagnetic materials, we need to evaluate the effective field. The effective field consists of severalterms, among which the demagnetizing field is of long-range nature. Evaluating the demagnetizing field directlyrequires work of O(N2) for a grid of N cells and thus it is the bottleneck in computational micromagnetics. Afast hierarchical algorithm using multipole approximation is developed to evaluate the demagnetizing field.

Wefirst construct a mesh hierarchy and divide the grid into boxes of different levels. The lowest level box is thewhole grid while the highest level boxes are just cells. The approximate field contribution from the cells containedin a box is characterized by the box attributes, which are obtained via multipole approximation.

The algorithmcomputes field contributions from remote cells using attributes of appropriate boxes containing those cells, and itcomputes contributions from adjacent cells directly.

Numerical results have shown that the algorithm requires workof O(NlogN) and at the same time it achieves high accuracy. It makes micromagnetic simulation in three dimensionsfeasible.