3.1 Magnetostatic field computation

We recall that magnetostatic field is defined by the Maxwell equations (1.45)-(1.46):

$$\begin{cases} \nabla\cdot{\mathbf{H}_\text{m}}=-\nabla\cdot\... ...abla\times{\mathbf{H}_\text{m}}=\mathbf{0} \end{cases} \quad,$$ (3.1)

with the following conditions at the body discontinuity surface $\partial \Omega$

$$\begin{cases} \textbf{n}\cdot\left[{\mathbf{H}_\text{m}}\rig... ...xt{m}}\right]_{\partial \Omega}= \mathbf{0} \end{cases} \quad.$$ (3.2)

From the simple inspection of the above equations, two important considerations can be drawn:

1. There is a functional relationship between magnetization $\textbf{M}$ and magnetostatic field ${\mathbf{H}_\text{m}}$, that is, the value of ${\mathbf{H}_\text{m}}$ at a spatial location $\textbf{r}$ depends on the value of magnetization vector field at every location $\textbf{r}'$ within $\Omega$. This is a consequence of the nonlocal (long range) character of magnetostatic maxwellian interactions. Each elementary dipole in the body contributes to produce the magnetostatic field at an assigned location within the body. Quoting Aharoni, ``...In a numerical computation of $N$ unit cells, the long range means the magnetostatic energy term includes an interaction of every cell with all the other cells, thus involving $N^2$ terms, whereas only $N$ terms are required for computing the other energy terms. Therefore, computing the magnetostatic energy takes almost all the computer time in a typical micromagnetic computation. It is also the energy term with the heaviest demand on the computer memory, which means that it determines the limit of the size of the body that a computer can handle...''.
2. The differential problem (3.1)-(3.2) is an open boundary problem, that is, even if one is interested in the computation of magnetostatic field ${\mathbf{H}_\text{m}}$ at locations $\textbf{r}\in \Omega$, one has to solve Eqs. (3.1)-(3.2) in the whole space. This means that, in the framework of numerical modeling, one should preform in principle the discretization of the whole space, which is, of course, not feasible. For this reason, numerical methods consistent with the continuum model have to be used in numerical simulations.

In this respect, the methods based on truncation of the interaction range of magnetostatic fields [61], mean-field approximation for distant particles [62], hierarchical dipole interaction evaluation schemes [63] cause loss of accuracy and indeed don't save very much computational time. In general, all the methods for the computation of the demagnetizing field have a cost scaling function that is something in between the minimum $\mathcal{O}(N)$ and the maximum $\mathcal{O}(N^2)$. In the sequel, we will analyze two different methods which are commonly used for magnetostatic field computations respectively in case of finite-differences and finite elements spatial discretization [52].