Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 341 / 3 / c / Solution

Extend a grid vector $v$ by zero to the Dirichlet boundary. Pairing contributions along undirected nearest-neighbour edges gives the discrete energy identitywhere $E$ includes edges from an interior node to a boundary node. This is the three-dimensional version of summation by parts for the seven-point Dirichlet Laplacian.

The right side is nonpositive. If it vanishes, every pair of neighbouring values agrees; connectivity of the grid and the zero boundary values then imply $v=0$. Thus $v^{T}Av<0$ for every nonzero $v$, so $A$ is negative definite. In particular, zero is not an eigenvalue, and therefore