Periodic Poisson equation on a four-point cycle ($2x_n-x_{n-1}-x_{n+1}=b_n$)

Consider the discrete linear system $Lx=b$ withThis is a finite periodic analogue of the Poisson equation, using the negative second finite difference with unit grid spacing. It is also a Graph Laplacian equation on a cycle graph; no continuum approximation is being claimed here.

Summing the four equations gives zero on the left, since each value occurs twice positively and twice negatively. A necessary condition is therefore ${\sum }_n{b}_n=0$. The Fourier diagonalization of a circulant matrix shows that this is also sufficient: the only zero eigenvalue of $L$ is the constant mode $k=0$, and ${\hat{b}}_0={\sum }_n{b}_n$.

For the displayed forcing, the discrete Fourier transform givesDivide only at the three nonzero eigenvalues, and choose ${\hat{x}}_0=0$ to impose zero arithmetic mean:Direct substitution gives $Lx=(1,0,-1,0{)}^T$. All other vectors satisfying the linear system are $x+a(1,1,1,1{)}^T$, because the kernel is the space of constant vectors. Thus balanced forcing determines the differences between values, while a reference level is an extra choice. This small model explains why a periodic Poisson equation needs both a compatibility condition and a normalization.