H-Matrix Techniques for Stray-Field Computations in Computational Micromagnetics

A major task in the simulation of micromagnetic phenomena is the effective computation of the stray-field H and/or of the corresponding energy, where H solves the magnetostatic Maxwell equations in the entire space. For a given FE magnetization mh, the naive computation of H via a closed formula typically leads to dense matrices and quadratic complexity with respect to the number N of elements. To reduce the computational cost, it is proposed to apply H-matrix techniques instead. This approach allows for the computation (and evaluation) of H in linear complexity even on adaptively generated (or unstructured) meshes. 1 Basic Micromagnetics Let Ω ⊂ R be the bounded spatial domain of a ferromagnet. Then, the magnetization m : Ω → R induces the so-called stray-field [9] (or demagnetization field)H : R → R, which is the solution of the magnetostatic Maxwell equations curlH = 0 and divB = 0 on R. (1) Here, B = H+m denotes the magnetic induction, with m extended by zero to R \Ω. Stokes’ Theorem implies H = −∇u, with a potential u that solves div (−∇u+m) = 0 in D′(Rd). (2)