The computed voltage from the edge function  variant of a separation of variable type solution V (resp. V)for the lower(resp. upper) region of the flow

matches the prescribed voltage of 1 (resp. 0) to not less than 10 places at points on the lower(resp. upper) contact. The corresponding current vectors in the lower(resp. upper) are the corresponding voltage gradients ∇V (resp. ∇V). The component of the current at points on the base layer segments 2,6,8,9 (resp. 5) in the horizontal(resp. vertical) direction match the prescribed 0 to 10 places.

The voltage(resp. vertical current component) computed from V and V match to not less than 10 places at all points on segment 3 or  4 to fulfill the standard continuity requirement across internal boundaries.

The current vector at all points on the entering(resp. exiting) segment 2(resp. 5) is in the vertical(resp. horizontal) direction.  The current magnitude at the reentrant becomes infinite at the reentrant

The idea of this being a simple solution is quickly dispelled by plotting the current component in the negative vertical direction along the lower boundary of the upper region

The singularity at the end of segment 4 is to be expected as Laplace’s equation has singular solutions of the form $r^n\cos \left(n\theta \right)$ with a gradient component $-n{r}^{n-1}\sin \left(n\theta \right)$ in the θ-direction that vanishes along the reentrant exiting segment 5 because θ is 0 and also along the entering segment 4 for n=k π/α , α=3 π/2,k∈K and K={1,2,4,5,7,8,…}.  The voltage distribution

$\sum _{k\in K}{a}_k{r}^{\frac{2k}{3}}\cos \left(\frac{2\theta k}{3}\right)$

(resp. current θ-component

$-\sum _{k\in K}{a}_k\frac{2}{3}k{r}^{\frac{2k}{3}-1}\sin \left(\frac{2\theta k}{3}\right)$

)has a finite(resp. infinite)value as r↓0 due to the factor $r^{2/3}$ (resp. $r^{-1/3}$)for all 0<θ<α   when the first superposition constant $a_1$ is non-zero. Non-zero  superposition constants in $a_2,a_3,\dots$ introduce infinite values in the corresponding higher order r-derivatives.

Since current vectors are normal to contacts, the 90° corners at the bottom(resp. top) of the lower(resp. upper)region do not force an abrupt change in the current direction thereby avoiding the presence of singularities.

The abrupt change in current direction at the rightmost lower corner of the upper region causes a mild discontinuity in the slope of the density

In an obituary to the originator  it was noted that the great advantage of the edge function method was the ease with which singularities, which arise from corners, cracks,inhomogeneities and reentrant angles could be incorporated into his scheme.

The reduced solution consisting of the solution less its singular part (1) has a smoothly varying vertical current component on the internal interface,

a similarly smoothly varying horizontal  current component

an undisturbed horizontal current along segment 2 due to the vertex function construction and a similarly smoothly varying voltage distribution along segment 1

paving the way to accurate low order separation of variable type representations in each region as a combination of explicit trigonometric×exponential functions from each side and explicit harmonic polynomials from the centroid. The coefficients in the combination are determined from the prescribed boundary conditions and the continuity requirements across the internal sub-region boundaries.