4.2.1 Preconditioning

Preconditioning of the system of linear equations involved in the Newton iteration of the Krylov subspace method can considerably speed up its solution [53]. In addition, this method leads to fewer function evaluations of the Landau-Lifshitz-Gilbert equation and allows larger time steps, which gives an excellent performance of the numerical time integration.

In order to find the root of Eq. (4.10), the Newton method requires the calculation of the intermediate corrections $\Delta \boldsymbol{y} = \boldsymbol{y}_{m} - \boldsymbol{y}_{m-1}$, which follow from

$$\frac{\partial \boldsymbol{G}}{\partial \boldsymbol{y}} \De... ...\boldsymbol{y} = -\boldsymbol{G}(\boldsymbol{y}_{m-1}) \quad.$$ (4.11)

The matrix $\partial \boldsymbol{G}/\partial \boldsymbol{y}$ in this linear system of equations is approximated by

$$\partial \boldsymbol{G}/\partial \boldsymbol{y} \approx I -... ...\frac{\partial \boldsymbol{f}}{\partial \boldsymbol{y}} \quad.$$ (4.12)

The calculation of the Jacobian of $\boldsymbol{f}$

$$\frac{\partial \boldsymbol{f}}{\partial \boldsymbol{y}}= \... ...ymbol{J} \times(\boldsymbol{J} \times \boldsymbol{H}) \right)$$ (4.13)

requires the calculation of the Jacobian of the total energy with respect to the magnetization.

We have derived the expressions for the gradient of the total energy in Sec. 3.3. Since the total energy is a simple sum of exchange, magnetocrystalline anisotropy, Zeeman, and magnetostatic energy, we have calculated their gradients individually. For the first two contributions we found, that the gradient of the energy is a linear function of magnetization and ended up with a matrix-vector formulation. The external field is explicitly given anyway, but the magnetostatic field had to be calculated with a hybrid finite element/boundary element method.

Now we can analyze their contributions to the Jacobian of the total energy. The external field does not contribute at all, since it is independent of the magnetization and its first derivative with respect to the magnetization is zero. The first derivative of the demagnetizing field would contribute. However, it is not considered for the calculation of the Jacobian for two reasons. First, its calculation would be very expensive in terms of computational effort, and, due to its long-range nature, it would lead to a full matrix for the Jacobian. This results in huge memory requirements and a lot of communication between the processors in a parallel program. Moreover, we do not need the exact Jacobian, but a sensible approximation, which still speeds up the Newton iterations. Thus, it is sensible to consider only the contributions from the exchange and magnetocrystalline anisotropy energy.

The calculation of the Jacobian of these two energy terms is finally very easy. We have already calculated their gradient with respect to the magnetization in order to calculate their contributions to the local field. As mentioned before, we found the energy gradients to be linear with respect to the magnetization and came up with a matrix-vector formulation. Due to this linearity, their Jacobians are just simply given by these matrices Eq. (3.21) and Eq. (3.28) and we just have to add them up to get the approximate Jacobian for the total energy.

Finally, instead of calculating Eq. (4.11) with Eq. (4.12) directly, the preconditioning technique is applied [54]: The linear system

$$A \boldsymbol{x} = \boldsymbol{b}$$ (4.14)

is rewritten as

$$(A P^{-1}) (P \boldsymbol{x}) = \boldsymbol{b}$$ (4.15)

and

$$A' \boldsymbol{x}' = \boldsymbol{b}$$ (4.16)

with $A'=A P^{-1}$ and $\boldsymbol{x}'=P \boldsymbol{x}$. If $P$ is a good approximation to $A$, then $A'$ is close to the identity matrix and Eq. (4.16) can be solved very efficiently.