Computational micromagnetics in three dimensions is of increasing interest with the development of magnetostrictive sensors andactuators. In solving the Landau-Lifshitz-Gilbert (LLG) equation, the governing equation of magnetic dynamics for ferromagnetic materials, we need to evaluate the effective field. The effective field consists of several terms, among which the demagnetizing field is of long-range nature.

Evaluating the demagnetizing field directly requires work of O(N^2) for a grid of N cells and thus it is the bottleneck in computational micromagnetics. A fast hierarchical algorithm using multipole approximation is developed to evaluate the demagnetizing field. We first construct a mesh hierarchy and divide the grid into boxes of different levels. The lowest level box is the whole grid while the highest level boxes are just cells. The approximate field contribution from the cells contained in a box is characterized by the box attributes, which are obtained via multipole approximation. The algorithm computes field contributions from remote cells using attributes of appropriate boxes containing those cells, and it computes contributions from adjacent cells directly. Numerical results have shown that the algorithm requires work of O(NlogN) and at the same time it achieves high accuracy. It makes micromagnetic simulation in three dimensions feasible.