Computing the electric potential generated by a current density across media with different conductivities
After doing more digging, I think I have found that my original suspicions were correct. There is some non-linearity.

I created a 2d finite element method model that approximates my conditions at least qualitatively: 



The subplot on the right is a cross-section taken from the y-center of the heat maps. This modeling result is backed up by the theory described here.

I was assuming the simple case of $$ -c\nabla^2 u=f $$ which my system reduces to when the three layers are identical and c is scalar. But in my case I need, $$  -\nabla *(c\nabla u)=f $$ to account for the "scattered" terms. 

I do not yet completely understand  the equations covered in this document, but I believe they are to key to this problem.