B.2.1 For the Scalar Function

The Neumann boundary integral for $\psi$ from (5.75) is modified to read

$$\begin{split}\int_{\mathcal{C}_{N2}}\lambda_i\vec{n}\cdot\lef... ...\mathrm{d}s = & = \left[D\right]\left\{c\right\} \end{split}$$ (B.22)

$$D_{ij}^e = \mu\left(\int_{\mathcal{C}^e_k}\lambda_i^e \mathrm{d}... ...t)\vec{n}_k\cdot\vec{\nabla}\lambda_j^e, i\in[1;3], j\in[1;3], k\in[1;3].$$ (B.23)

For the computation of $\left[D\right]$ the following expressions are obtained:

$$\left(l_{12}\vec{n}_3\right)\cdot\vec{\nabla}\lambda_1^e = -\left... ...vec{r}_{23}\times\vec{e}_z}{2F_e} = -\frac{\vec{r}_{12}\cdot\vec{r}_{23}}{2F_e}$$ (B.24)

and analogously

$$\left(l_{12}\vec{n}_3\right)\cdot\vec{\nabla}\lambda_2^e = -\frac... ...ight)\cdot\vec{\nabla}\lambda_3^e = -\frac{\vec{r}_{12}\cdot\vec{r}_{12}}{2F_e}$$ (B.25)

$$\begin{split}& \left(l_{23}\vec{n}_1\right)\cdot\vec{\nabla}\... ...a_3^e = -\frac{\vec{r}_{23}\cdot\vec{r}_{12}}{2F_e} \end{split}$$ (B.26)

$$\begin{split}& \left(l_{31}\vec{n}_2\right)\cdot\vec{\nabla}\... ..._3^e = -\frac{\vec{r}_{31}\cdot\vec{r}_{12}}{2F_e}. \end{split}$$ (B.27)

Along Edge $12$ :

$$D_{1j}^e = \mu\left( \int_{\mathcal{C}^e_1}\lambda_1^e \mathrm{... ...bda_j^e = \frac{\mu}{2}\left(l_{12}\vec{n}_3\right)\cdot\vec{\nabla}\lambda_j^e$$ (B.28)

$$D_{2j}^e = \mu\left( \int_{\mathcal{C}^e_1}\lambda_2^e \mathrm{... ...bda_j^e = \frac{\mu}{2}\left(l_{12}\vec{n}_3\right)\cdot\vec{\nabla}\lambda_j^e$$ (B.29)

$$D_{3j}^e = \mu\left( \int_{\mathcal{C}^e_1}\lambda_3^e \mathrm{... ...3\cdot\vec{\nabla}\lambda_j^e = 0, \lambda_3^e = 0 \mathrm{on Edge} 12$$ (B.30)

$$\left[D\right]^e = -\frac{\mu}{4F_e} \left[ \begin{array}{ccc} \v... ...t\vec{r}_{31} & \vec{r}_{12}\cdot\vec{r}_{12} 0 & 0 & 0 \end{array} \right].$$ (B.31)

Analogously for the remaining edges the following is calculated.

Along Edge $23$ :

$$\left[D\right]^e = -\frac{\mu}{4F_e} \left[ \begin{array}{ccc} 0 ... ...c{r}_{23}\cdot\vec{r}_{31} & \vec{r}_{23}\cdot\vec{r}_{12} \end{array} \right].$$ (B.32)

Along Edge $31$ :

$$\left[D\right]^e = -\frac{\mu}{4F_e} \left[ \begin{array}{ccc} \v... ...c{r}_{31}\cdot\vec{r}_{31} & \vec{r}_{31}\cdot\vec{r}_{12} \end{array} \right].$$ (B.33)