Exercise 4.4.3

The Laplace with 3 variables is $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}u}{\partial z^2}=f(x,y,z)$

In 2D grid, we have $x,y$ discrete points on a $N\times N$ grid, and we need compute the derivatives on all the $(x_i,y_i)$ grid points w.r.t. $x$ and $y$ separately. So for $x$ component, we have $K_x=I_N\otimes D_N$, which has $N^2$ points. Also $K_y=D_N\otimes I_N$.

Now in 3D grid, we will have $x,y,z$ on $N^3$ discrete points. We need compute the derivatives w.r.t $x,y,z$ each on $N^3$ cubic grid points. Suppose we have the 2D matrix $K_x$, that is for one $z$ value, image we repeat this for $N$ of $z$ values, that requires a $I_N\otimes K_x=I_N\otimes I_N\otimes D_N$ matrix, which has $N^3$ dimension.

Similarly we find the $K_y,K_z$, so the final $K=K_x+K_y+K_z=I_N\otimes I_N\otimes D_N+I_N\otimes D_N\otimes I_N+D_N\otimes I_N\otimes I_N$