Laplaciano en cartesianas:
$latex Delta u = Sigma_i frac{partial^2}{partial x_i^2}u$
1d
$latex frac{u_{i-1}-2u_i+u_{i+1}}{h^2} = f_i $
$latex frac{1}{h^2}u_{i-1} + frac{1}{h^2}u_{i+1} +frac{-2}{h^2}u_i= f_i$
2d
$latex frac{u_{i-1,j}-2u_{i,j}+u_{i+1,j}}{h_x^2} + frac{u_{i,j-1}-2u_{i,j}+u_{i,j+1}}{h_y^2} = f_{i,j}$
$latex i$ fijo:
$latex frac{1}{h_y^2}u_{i,j-1} + frac{1}{h_y^2}u_{i,j+1} +(frac{-2}{h_x^2}+frac{-2}{h_y^2})u_{i,j}= g_{i,j}(:=f_{i,j} + frac{-1}{h_x^2}u_{i-1,j} + frac{-1}{h_x^2}u_{i+1,j})$
$latex j$ fijo:
$latex frac{1}{h_x^2}u_{i-1,j} + frac{1}{h_x^2}u_{i+1,j} +(frac{-2}{h_x^2}+frac{-2}{h_y^2})u_{i,j}= g_{i,j}(:=f_{i,j} + frac{-1}{h_y^2}u_{i,j-1} + frac{-1}{h_y^2}u_{i,j+1})$
3d
$latex frac{u_{i-1,j,k}-2u_{i,j,k}+u_{i+1,j,k}}{h_x^2} + frac{u_{i,j-1,k}-2u_{i,j,k}+u_{i,j+1,k}}{h_y^2} + frac{u_{i,j,k-1}-2u_{i,j,k}+u_{i,j,k+1}}{h_z^2} = f_{i,j,k}$
$latex i,j$ fijos:
$latex frac{1}{h_z^2}u_{i,j,k-1} + frac{1}{h_z^2}u_{i,j,k+1} +(frac{-2}{h_x^2}+frac{-2}{h_y^2}+frac{-2}{h_z^2})u_{i,j,k}= g_{i,j,k}$
$latex (g_{i,j,k}:=f_{i,j,k} + frac{-1}{h_x^2}u_{i-1,j,k} + frac{-1}{h_x^2}u_{i+1,j,k} + frac{-1}{h_y^2}u_{i,j-1,k} + frac{-1}{h_y^2}u_{i,j+1,k})$
$latex i,k$ fijos:
$latex frac{1}{h_y^2}u_{i,j-1,k} + frac{1}{h_y^2}u_{i,j+1,k} +(frac{-2}{h_x^2}+frac{-2}{h_y^2}+frac{-2}{h_z^2})u_{i,j,k}= g_{i,j,k}$
$latex (g_{i,j,k}:=f_{i,j,k} + frac{-1}{h_x^2}u_{i-1,j,k} + frac{-1}{h_x^2}u_{i+1,j,k} + frac{-1}{h_z^2}u_{i,j,k-1} + frac{-1}{h_z^2}u_{i,j,k+1})$
$latex j,k$ fijos:
$latex frac{1}{h_x^2}u_{i-1,j,k} + frac{1}{h_x^2}u_{i+1,j,k} +(frac{-2}{h_x^2}+frac{-2}{h_y^2}+frac{-2}{h_z^2})u_{i,j,k}= g_{i,j,k}$
$latex (g_{i,j,k}:=f_{i,j,k} + frac{-1}{h_y^2}u_{i,j-1,k} + frac{-1}{h_y^2}u_{i,j+1,k} + frac{-1}{h_z^2}u_{i,j,k-1} + frac{-1}{h_z^2}u_{i,j,k+1})$