I am working on fast Poisson solvers and I have to understand some basic concepts with the discretisation of the 2D Poisson equation ($U_{xx}+U_{yy}=f$) in the Sylvester equation form ($KX+XK=F$).
I have been able to discretise the 2D Poisson equation using the central difference approximation (assuming $\Delta x=\Delta y=h$) and I got $$-\dfrac{1}{h^{2}}\left[ 2U_{i,j}-U_{i-1,j}-U_{i+1,j}\right]-\dfrac{1}{h^{2}}\left[ 2U_{i,j}-U_{i,j-1}-U_{i,j+1}\right]=f_{i,j}$$ Now, instead of simplifying this to get a linear equation $A\vec{u}=\vec{f}$ where $A$ is an $(n-1)(n-1)\times (n-1)(n-1)$ matrix and $\vec{u}, \vec{f}$ are column vectors. I need to express the discretised 2D Poisson equation in the Sylvester equation form $KX+XK=F$ where $$K=-\dfrac{1}{h^{2}}\begin{bmatrix} 2&-1&0&0&0&\ldots&0\\ -1&2&-1&0&0&\ldots&0\\ 0&-1&2&-1&0&\ldots&0\\ \vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\ 0&\ldots&0&-1&2&-1&0\\ 0&\ldots&\ldots&0&-1&2&-1\\ 0&\ldots&\ldots&\ldots&0&-1&2\\ \end{bmatrix}$$ and matrix $X$ represents the values of the solution on the interior nodes of a $(n+1)\times (n+1)$ equispaced grid.
I finding it difficult to find the transition from the discretised 2D Poisson equation to the Sylvester equation. Please I need help from whoever has idea about this.
The linear system can also be denoted as follows. Let L be the 1D Laplacian matrix (tridiagonal), then the 2D equation becomes:
Kron(I, L)+kron(L, I)u=f
If we write u f as matrices U and F (rows correspond to y coordinates and columns to x coordinates), then we have LU + UL = F This is called a Lyapunov/Sylvester equation.