Simplify Laplace equation in rectangle geometry

A numerical solution of the Laplace equation (in a rectangular geometry) can be obtained with an equivalent resistor network. Key internet references are found here: In our case, the domain of interest in subdivided in rectangles:
Let the width and height of a rectangle be given by $\,dx\,$ and $\,dy\,$ respectively, then each of the four edges is associated with a resistor $\,R\,$ having admittance $\,A_x = \lambda/dx\cdot dy/2\,$ for the horizontal edges and $\,A_y = \lambda/dy\cdot dx/2\,$ for the vertical edges, where $\lambda$ is the conductivity (equal to $a_1$ or $a_2$ in the OP's question). Resulting in the following "Finite Element matrix" for one rectangle: $$ \begin{bmatrix} A_x+A_y & - A_x & - A_y & 0 \\ - A_x & A_x+A_y & 0 & - A_y \\ - A_y & 0 & A_x+A_y & - A_x \\ 0 & - A_y & - A_x & A_x+A_y \end{bmatrix} $$ The rest of the numerical treatment is pretty standard Finite Element methodology: 
program hesam;

Uses Laplace;

procedure test;
var
  k : integer;
begin
  Starten; {Initialize }
  for k := 0 to 9 do
  begin
  { FEM Calculations }
    Rekenen(Random,Random,0,1);
  { Store in 'results' file }
    Opschrijven(k);
  end;
end;

begin
  test;
end.
Here is a link to the complete (Delphi Pascal) unit that does the FEM Calculations: Below is the $\,40\times 30\,$ grid that has been used for sample calculations and a contour map of some results with $V_0=0$ , $V_1=1$ and random $a_1,a_2$.

Sample output ('result' file) - can you see where it is? - : 

   x   y       V[x,y]
  20   0  5.00000000000000E-0001
  20   1  5.00000000000000E-0001
  20   2  5.00000000000000E-0001
  20   3  4.99999999999999E-0001
  20   4  4.99999999999999E-0001
  20   5  4.99999999999999E-0001
  20   6  4.99999999999999E-0001
  20   7  4.99999999999999E-0001
  20   8  4.99999999999999E-0001
  20   9  5.00000000000000E-0001
  20  10  5.00000000000001E-0001
  20  11  5.00000000000001E-0001
  20  12  5.00000000000001E-0001
  20  13  5.00000000000002E-0001
  20  14  5.00000000000002E-0001
  20  15  5.00000000000002E-0001
  20  16  5.00000000000003E-0001
  20  17  5.00000000000003E-0001
  20  18  5.00000000000003E-0001
  20  19  5.00000000000002E-0001
  20  20  5.00000000000002E-0001
  20  21  5.00000000000003E-0001
  20  22  5.00000000000002E-0001
  20  23  5.00000000000002E-0001
  20  24  5.00000000000001E-0001
  20  25  5.00000000000001E-0001
  20  26  5.00000000000001E-0001
  20  27  5.00000000000001E-0001
  20  28  5.00000000000000E-0001
  20  29  5.00000000000000E-0001
  20  30  5.00000000000000E-0001