4.3.1 Discretized free energy and effective field

Usual spatial discretization techniques [52] (e.g. finite elements and finite differences) quite naturally lead to a discretized version of the free energy (4.3) which has generally the form

$$\underline{{\text{g}}}(\underline{\textbf{m}},\underline{\textbf{... ...rline{\textbf{m}}- \underline{\textbf{h}}_a \cdot \underline{\textbf{m}}\quad.$$ (4.19)

where ${\underline{\textrm{C}}}$ is now a $3N\times 3N$ symmetric matrix [80] which describes exchange, anisotropy and magnetostatic interactions. Once the free energy has been discretized, the corresponding spatially discretized effective field $\underline{\textbf{h}}_$eff$(\underline{\textbf{m}},t)$ can be obtained as

$$\underline{\textbf{h}}_ (\underline{\textbf{m}},t)=-\frac{\partial \underline{{\text{g}}}... ...rline{\textrm{C}}}\cdot\underline{\textbf{m}}+\underline{\textbf{h}}_a(t) .$$ (4.20)

We notice that the effective field mathematical structure (4.12) is formally preserved after the spatial discretization, and the matrix ${\underline{\textrm{C}}}$ is the discretized version of the formally self-adjoint integro-differential operator $\mathcal{C}$. The matrix ${\underline{\textrm{C}}}$ can be naturally decomposed into the sum of the three terms ${\underline{\textrm{C}}}_$ex, ${\underline{\textrm{C}}}_$m, ${\underline{\textrm{C}}}_$an which correspond to discretized exchange, magnetostatic and anisotropy interactions:

$${\underline{\textrm{C}}}={\underline{\textrm{C}}}_ +{\underline{\textrm{C}}}_ +{\underline{\textrm{C}}}_ \quad.$$ (4.21)

It is important to observe that ${\underline{\textrm{C}}}_$ex and ${\underline{\textrm{C}}}_$an are sparse matrices since the exchange and anisotropy interactions have a local character, whereas ${\underline{\textrm{C}}}_$m is a full matrix owing to the long-range magnetostatic interactions.