Appendix: Weak Formulation¶

Strong formulation¶

In the collector: $i_1 = -\sigma \nabla \Phi_1$ $\nabla \cdot i_1 = 0$
In the electrode: $i_1 = -\sigma \nabla \Phi_1$ $i_2 = -\kappa \nabla \Phi_2$ $\nabla \cdot i_1 = \nabla \cdot i_2 = a i_n$
In the seperator: $i_2 = -\kappa \nabla \Phi_2$ $\nabla \cdot i_2 = 0$

Weak formulation¶

In the collector:

$-\int_{\Omega_c}dr \phi_{1,i} \sigma \Delta \Phi_{1,j}\phi_{1,j} = -\int_{\partial {\Omega_c}}dr \phi_{1,i} \sigma \nabla \Phi_{1,j}\phi_{1,j} + \int_{\Omega_c}dr \nabla \phi_{1,i} \sigma \nabla \Phi_{1,j}\phi_{1,j}$

In the electrode:

$\begin{pmatrix} \int_{\Omega_e}dr \phi_{1,i} (-\sigma) \Delta \Phi_{1,j}\phi_{1,j} + a C \frac{\partial \Phi_{1,j}\phi_{1,j}}{\partial t} & - \int_{\Omega_e}dr \phi_{1,i} aC \frac{\partial \Phi_{2,j}\phi_{2,j}}{\partial t} \\ -\int_{\Omega_e} dr \phi_{2,i} ac \frac{\partial \phi_{1,j}}{\partial t} & \int_{\Omega_2} dr \phi_{2,i} (-\kappa) \Delta \Phi_{2,j} \phi_{2,j} - \phi_{2,i} aC \frac{\partial \Phi_{2,j}\phi_{2,j}}{\partial t} \end{pmatrix} = \begin{pmatrix} -\int_{\partial \Omega_e} dr \phi_{1,i} \ sigma \nabla \Phi_{1,j}\phi_{1,j} + \int_{\Omega_e}dr \nabla \phi_{1,i} \sigma) \nabla \Phi_{1,j}\phi_{1,j} + a C \frac{\partial \Phi_{1,j}\phi_{1,j}}{\partial t} & - \int_{\Omega_e}dr \phi_{1,i} aC \frac{\partial \Phi_{2,j}\phi_{2,j}}{\partial t} \\ -\int_{\Omega_e} dr \phi_{2,i} aC \frac{\partial \phi_{1,j}}{\partial t} & -\int_{\partial \omega_e} dr \phi_{2,i}\kappa \nabla \Phi_{2,j} \phi_{2,j} + \int_{\Omega_e} dr \nabla \phi_{2,i} \kappa \nabla \Phi_{2,j} \phi_{2,j} - \phi_{2,i} aC \frac{\partial \Phi_{2,j}\phi_{2,j}}{\partial t} \end{pmatrix}$

In the seperator:

$-\int_{\Omega_s}dr \phi_{2,i} \kappa \Delta \Phi_{2,j}\phi_{2,j} = -\int_{\partial {\Omega_s}}dr \phi_{2,i} \kappa \nabla \Phi_{2,j}\phi_{2,j} + \int_{\Omega_s}dr \nabla \phi_{2,i} \sigma \nabla \Phi_{2,j}\phi_{2,j}`$