Numerical Algorithms in Computational Micromagnetics

The magnetization of a ferromagnet can be described by the phenomenological equation due to Landau, Lifshitz, and Gilbert (LLG). This well-established model accounts to effects of magnetic exchange interaction, anisotropy, demagnetization and exterior fields on the basis of the Landau-Lifshitz free energy, and is widely accepted as the relevant model to describe dynamical micromagnetism. Numerical micromagnetism is an active area of research, with the goal being the design of reliable, stable schemes to compute magnetic microstructures or averages thereof. Most recent mathematical research concerned stationary micromagnetics while first mathematical studies for (LLG) are only very recent and demanding further development.This is in contrast to a well-established scientific community of computational electromagnetism where numerical experience is mainly based on an empirical level. The project aims to contribute to the mathematical foundations of those simulations, important for modern technologies. It is based on our mathematical experience on the stationary situation and the time-depending problems, as well as the numerical experience of our cooperation partner that we address three goals in this project: (i) Construction and analysis of stable, efficient time-space discretization schemes to numerically solve (LLG) in the case of vanishing exchange interaction. (ii) Study of level set methods for solving `solitonian equations` for the stationary problem in order to recover microscopic information out of calculated macroscopic magnetizations. (iii) Construction and analysis of time-space adaptivity for (LLG) in a master example in 1D.

The understanding of micromagnetic phenomena is of great importance for the development and improvement of modern storage devices, e.g. hard discs, or even for magnetic sensors, e.g. fingerprint sensors. As the development of a new hardware is rather costly, it usually pays to do some numerical simulations before. The project "Reliable and Efficient Numerical Algorithms in Computational Micromagnetics" funded by the FWF under grant P15274-N12 was concerned with the mathematical foundation and the development of numerical tools for micromagnetic simulations: An algorithm is reliable if it includes some rigorous mathematical error control for the quantity of interest. For instance, in the theory of micromagnetics one is interested in the computation of the so-called magnetization field m which lives on a ferromagnetic body and which characterizes its magnetic properties. A numerical simulation provides a discrete solution m h which depends on a triangulation of the ferromagnetic body. Therefore, one task of the project was the de-velopment of adaptive algorithms which steer the mesh-refinement automatically and simultaneously provide an error control of m - m h . These algorithms allow the computation of an approximation m h of m up to a given tolerance. They are based on the mathematical derivation of so-called a posteriori error estimates, where the upper bound of the error m - m h depends only on computable quantities, i.e. the upper bound does not depend on the (in general unknown) magnetization field m. One numerically challenging point in micromagnetic simulations is the necessary computation of the so-called demagnetization field H which is self-induced by the magnetization m. Let`s assume that the triangulation of the ferromagnetic body consists of N elements. Then, the naive computation of the demagnetization field H h corresponding to the discrete magnetization m h leads to N2 arithmetic operations as well as quadratic storage requirements. This means that if you refine your triangulation and double the number of elements to get a better approximation of m, the computation of m h takes four times longer and needs four times more memory. Therefore micromagnetic simulations were often restricted by the computational time as well as by the storage requirements in the past. However, this bottleneck can be overcome: Recently introduced data compression techniques by so- called hierarchical matrices can be applied to compress the storage in such a way that the computation of H h and the corresponding storage requirements depend only linearly on N instead of quadratically. Obviously, this provides and allows much more efficient and much larger simulations. Moreover, the error of the data compression can be controlled so that our algorithms don`t loose their reliability. The numerical evidence raised within the project clearly predicts that the hierarchical matrix techniques developed for this application will become soon the most important tool in electromagnetism.