4.3 Spatially semi-discretized LLG equation

We now introduce a spatially discretized version of the mathematical model. The discussion presented below is considerably general and thus applicable to all the usual spatial discretization techniques [52]. To start the discussion, let us assume that the magnetic body has been subdivided in $N$ cells or finite elements. We denote the magnetization vector associated to the $l$-th cell or node by $\textbf{{m}}_l(t)\in \mathbb{R}^3$. Analogously, the effective and the applied fields at each cell will be denoted by the vector $\textbf{h}_{\text{eff},l}(t)$, $\textbf{h}_{a,l}(t)$. In addition to the cell-vectors, we introduce another notation for the mesh vectors which include the information of all cells of the mesh. In this respect, we will indicate with $\underline{{\textbf{m}}}$, $\underline{\textbf{h}}_$eff, $\underline{\textbf{h}}_$a the vectors in $\mathbb{R}^{3N}$ given by:

$$\underline{\textbf{m}}=\left( \begin{array}{c} \textbf{{m}}_1 \vdots \textbf{{m}}_N \end{array} \right) \qquad \underline{\textbf{h}}_ =\left( \begin{array}{c} \textbf{h}_\text{eff,1} \vdots \textbf{h}_\text{eff,N} \end{array} \right) \qquad \underline{\textbf{h}}_ =\left( \begin{array}{c} \textbf{h}_\text{a,1} \vdots \textbf{h}_\text{a,N} \end{array} \right) \quad.$$ (4.18)