Laplace Equation and Square Sums

Clear["`*"]

1)  Form an accurate polynomisl approximation to ϕ[x,y] near [0,0]

ϕa[x_, y_]  = 
      Expand[Series[ϕ[x, y], {x, 0, 6}, {y, 0, 6}] // Normal]

2)  Look at the sum of nearest neighbors

Sh  =  
      Expand[(ϕa[h, 0] + ϕa[-h, 0] + ϕa[0, h] + ϕa[
                0, -h])/4]

3)  Look at the sum of next-nearest neighbors

Sd  =  
      Expand[(ϕa[h, h] + ϕa[-h, h] + ϕa[
                h, -h] + ϕa[-h, -h])/4]

4) Look at the "magic" average

S  =  Expand[(4 Sh + Sd)/5]

5) Rewrite in terms of Laplace operator Δ.

Sa = Series[S/. Derivative[0, 2][ϕ][0, 0]->- Derivative[2, 0][ϕ][0, 0] + Δϕ[0, 0]/. Derivative[0, 4][ϕ][0, 0]-> -2 Derivative[2, 2][ϕ][0, 0]- Derivative[4, 0][ϕ][0, 0]+ΔΔϕ[0, 0],{h, 0, 6}]