Study of the Discretisation of the Ionic Mass Transport Equations with the “Inverse Scharfetter-Gummel” Method

In order to describe an electrochemical system correctly with finite element simulations, close to the electrodes a fine grid is needed (10 times more refined than close to the boundaries for fluid flow calculations), because there the gradients of the potential and the concentrations are large and far from linear.
Simulations of these systems are very time and memory consuming. In a search for a better description of diffusion and migration dominated transport with less points in the grid, which means a decrease of the allocated memory, the "Inverse Scharfetter Gummel" discretisation was studied.
Instead of using linear shape functions for the discretisation of the potential field, the current density in an element is assumed to be constant. Together with the linear shape functions for the concentration fields this leads to logarithmic shape functions for the potential. This technique was derived from the well known Scharfetter-Gummel method widely employed in finite element simulations of drift-diffusion problems in semiconductor device modelling.
A comparison of results obtained from simulations with linear shape functions and logarithmic shape functions for the potential will be presented for 1D as well as for 2D geometries.