The integrals of

\(-\varepsilon \nabla \psi \cdot \nu\) and

\(-\lambda \nabla T \cdot \nu\) over the interface

\(\partial V_r \cap \partial V_l\) are approximated by the conventional finite difference approximations

$$\begin{aligned} \int \limits _{\partial V_r \cap \partial V_l} -\varepsilon \nabla \psi \cdot \nu \,{\text {d}}\Gamma&\approx \frac{{\mathrm {mes}}\left( \partial V_r \cap \partial V_l\right) }{\left| x_l - x_r\right| } \varepsilon \left( \psi _l - \psi _r\right) , \\ \int \limits _{\partial V_r \cap \partial V_l} -\lambda \nabla T \cdot \nu \,{\text {d}}\Gamma&\approx \frac{{\mathrm {mes}}\left( \partial V_r \cap \partial V_l\right) }{\left| x_l - x_r\right| } \lambda \left( T_l - T_r\right) , \nonumber \end{aligned}$$

(10)

where

\({\mathrm {mes}}(K)\) denotes the measure of a set

K. Meanwhile, the corresponding integrals in the continuity equations are approximated with some extra effort

$$\begin{aligned} \begin{aligned} \int \limits _{\partial V_r \cap \partial V_l} j_n \cdot \nu \,{\text {d}}\Gamma&\approx \frac{{\mathrm {mes}}\left( \partial V_r \cap \partial V_l\right) }{\left| x_l - x_r\right| } J_n^{l; r},\\ \int \limits _{\partial V_r \cap \partial V_l} j_p \cdot \nu \,{\text {d}}\Gamma&\approx \frac{{\mathrm {mes}}\left( \partial V_r \cap \partial V_l\right) }{\left| x_l - x_r\right| } J_p^{l; r}, \end{aligned} \end{aligned}$$

where the numerical fluxes

\(J_n^{l; r}\) and

\(J_p^{l; r}\) are determined by a modification of the Scharfetter–Gummel scheme based on averaging of inverse activity coefficients introduced in [

18] and discussed with respect to degenerate semiconductors in [

9,

19]. We introduce some notation to define the expressions for

\(J_n^{l; r}\) and

\(J_p^{l; r}\) in (

12), (

13). Let

$$\begin{aligned} \psi _{l,r}:= & {} \frac{\psi _l + \psi _r}{2}, \ \varphi _{n; l,r} := \frac{\varphi _{n;l} + \varphi _{n;r}}{2}, \ \varphi _{p; l,r} := \frac{\varphi _{p;l} + \varphi _{p;r}}{2}, \ T_{l,r} := \frac{T_l + T_r}{2}, \\ \overline{\eta }_{n;l}:= & {} \eta _n\left( \psi _l, \varphi _{n;l}, T_{l,r}\right) , \ \overline{\eta }_{n;r} := \eta _n\left( \psi _r, \varphi _{n;r}, T_{l,r}\right) , \ \overline{\eta }_n^{l,r} := \eta _n\left( \psi _{l,r}, \varphi _{n; l,r}, T_{l,r}\right) , \\ \overline{\eta }_{p;l}:= & {} \eta _p\left( \psi _l, \varphi _{p;l}, T_{l,r}\right) , \ \overline{\eta }_{p;r} := \eta _p\left( \psi _r, \varphi _{p;r}, T_{l,r}\right) , \ \overline{\eta }_p^{l,r} := \eta _p\left( \psi _{l,r}, \varphi _{p; l,r}, T_{l,r}\right) , \\ U_T^{l,r}:= & {} \frac{k_{\mathrm {B}} T_{l,r}}{q}, \ s_n^{l,r} := \frac{\sigma _n}{k_{\mathrm {B}} T_{l,r}}, \ s_p^{l,r} := \frac{\sigma _p}{k_{\mathrm {B}} T_{l,r}}, \\ n^{l,r}:= & {} N_{n0} G\left( \overline{\eta }_n^{l, r}; s_n^{l,r}\right) ,\ p^{l,r} := N_{p0} G\left( \overline{\eta }_p^{l, r}; s_p^{l,r}\right) ,\\ \mu _n^{l,r}:= & {} \mu _n\left( T_{l,r}, n^{l,r}, F^{l,r}\right) , \ \mu _p^{l,r} := \mu _p\left( T_{l,r}, p^{l,r}, F^{l,r}\right) , \end{aligned}$$

where

\(\eta _n\) and

\(\eta _p\) are defined in (

3). Note that according to (

4), the mobility over a surface

\(\partial V_l\cap \partial V_r\) depends on the modulus of the gradient

\(|\nabla \psi |\). The finite difference approximation behind the two point flux finite volume ansatz (

10) gives only the normal component of the gradient with respect to

\(\partial V_l\cap \partial V_r\) and misses the tangential contribution, allowing for weak convergence at best if scaled with the space dimension [

20]. Therefore, we compute the approximation of

\(|\nabla \psi |^{2}\) on

\(\partial V_l\cap \partial V_r\) as the average squared norms of the gradients of the P1 finite element reconstruction

\(\psi _\tau\) over the set

\(\mathcal T_{l,r}\) of all simplices

\(\tau\) (triangles in 2D) in the underlying Delaunay triangulation adjacent to the edge

\(\overline{x_lx_r}\) [

21]:

$$\begin{aligned} \big |\nabla \psi \big |^2 |_{\partial V_l\cap \partial V_r} \approx \frac{\sum _{\tau \in {\mathcal {T}}_{l,r}} {\mathrm {mes}}(\tau ) |\nabla \psi _\tau |^2}{\sum _{\tau \in {\mathcal {T}}_{l,r}} {\mathrm {mes}}(\tau )}=:(F^{l,r})^2. \end{aligned}$$

(11)

With the above definitions, the numerical fluxes

\(J_n^{l;r}\) and

\(J_p^{l;r}\) have the form

$$\begin{aligned} J_n^{l;r}= & {} - q N_{n0} \mu _n^{l,r} U_T^{l,r} \frac{G\left( \overline{\eta }_n^{l,r}; s_n^{l,r}\right) }{\exp \left( \overline{\eta }_n^{l,r}\right) } \left[ \exp \left( \overline{\eta }_{n; l} \right) B\left( \frac{\psi _l - \psi _r}{U_T^{l,r}}\right) - \exp \left( \overline{\eta }_{n; r}\right) B\left( -\frac{\psi _l - \psi _r}{U_T^{l,r}}\right) \right] , \end{aligned}$$

(12)

$$\begin{aligned} J_p^{l;r}= & {} q N_{p0} \mu _p^{l,r} U_T^{l,r} \frac{G\left( \overline{\eta }_p^{l,r}; s_p^{l,r}\right) }{\exp \left( \overline{\eta }_p^{l,r}\right) } \left[ \exp \left( \overline{\eta }_{p; l} \right) B\left( -\frac{\psi _l - \psi _r}{U_T^{l,r}}\right) - \exp \left( \overline{\eta }_{p; r}\right) B\left( \frac{\psi _l - \psi _r}{U_T^{l,r}}\right) \right] . \end{aligned}$$

(13)

Here,

B denotes the Bernoulli function,

\(B\left( x\right) = \frac{x}{\exp \left( x\right) - 1}\).