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Let $h=\Delta x$. A one-dimensional Taylor expansion givesAdding the three coordinate directions shows that the exact solution has stencil defectThis is the local defect of the seven-point Dirichlet Laplacian. Its inverse has max-norm size $O(h^{-2})$: this follows from the discrete maximum principle, for example by comparison with the grid restriction of $x_1(1-x_1)+x_2(1-x_2)+x_3(1-x_3)$, whose stencil is $-6h^2$. Therefore the grid error is $O(h^2)$.

The question writes this error as $O(h^{p+1})$, soIn the more usual terminology, the finite difference method is second-order accurate.

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