Five-point stiffness stencil on a diagonal triangular grid ($4U_{ij}-U_{i-1,j}-U_{i+1,j}-U_{i,j-1}-U_{i,j+1}$)

For conforming piecewise affine elements on uniform squares split along their positive diagonals, a lower-triangle element matrix is $\frac{1}{2}[[1,-1,0],[-1,2,-1],[0,-1,1]]$ in consecutive vertex order. Finite element assembly gives diagonal entry 4 and four axis-neighbor entries $-1$. The diagonal mesh edges contribute zero coupling because the corresponding local shape-function gradients are perpendicular. Strong Dirichlet elimination removes known-neighbor columns and adjusts the load.