I want to set up a surface charge boundary condition for the simulation of semiconductor and electrolyte interface.
The 2D-Poisson example and surface charge boundary condition are shown in following picture.
And then I substitute the 2nd part of LHS with boundary condition.
The final results are shown in figure 3.
After I finished the substitution of boundary condition. The 2nd part of Poisson's equation became a constant value which is the surface charge density divided by y-direction meshing. This result let me feel a little bit strange. The influence of y-direction flux is now disappeared!?
Is this result correct?
If it is not, which kind if boundary condition should I apply for the surface charge simulation?