5 Finite Element Scheme

$\Lambda_{h}(t_{n})$ is a triangulation of the area $T$ at discrete time $t_{n}$, and $\{\phi_{i}^{n-1}\}_{i=0}^{N-1}$ are discrete nodal values of the order parameter on this triangulation. A finite element based iteration for solving (4.50) on grid $\Lambda_{h}(t_{n})$ and evaluating the order parameter at the time $t_{n+1}=t_{n}+\Delta t$ consists of two steps [88]:

Step 1. For the $k^{th}$ iteration of the $n+1^{th}$ time step the linear system of equations has to be solved:

$$\epsilon \frac{\pi}{2} M_{ii}\phi_{i}^{n+1,k}+\Delta t D_{s}K_{ii}\mu_{i}^{n+1,k}=\alpha_{i}$$ (265)

$$M_{ii}\mu_{i}^{n+1,k}-\tau\biggl(\frac{1}{\epsilon}M_{ii}+\epsilon K_{ii}\biggr)\phi_{i}^{n+1,k}=\beta_{i},$$ (266)

where

$$\alpha_{i}=\epsilon \frac{\pi}{2}M_{ii}\phi_{i}^{n}-\Delta t D_{s}\sum_{i\neq j}K_{ij}\mu_{j}^{n+1,k-1}$$ (267)

$$\beta_{i}=\tau\epsilon \sum_{i\neq j} K_{ij}\phi_{j}^{n+1,k-1}-\vert e\vert Z^{*}M_{ii}V_{i}^{n}$$ (268)

for each $i=0,N-1$ of the nodal values $(\phi^{n+1}_{i},\mu^{n+1}_{i})$ of the triangulation $\Lambda_{h}(t_{n})$. $[M]_{ij}$ and $[K]_{ij}$ are the lumped mass and stiffness matrix, respectively and $\tau = \frac{4 \Omega \gamma_{s}}{\pi}$.

Step 2. All nodal values $\{\phi_{i}^{n+1}\}_{i=0}^{N-1}$ are projected on $[-1, 1]$ by a function

$$\rho(x)=max(-1,min(1,x)).$$ (269)

For solving (4.52) a conventional finite element scheme is applied [76].