Five-point Laplacian ($h^{-2}{\Gamma }_5$)

On a square mesh, the dimensionless five-point Laplacian is ${\Gamma }_{5}u=u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i,j-1}-4u_{i,j}$. The scaled operator $h^{-2}{\Gamma }_5$ approximates the Laplacian with error $O(h^2)$ for sufficiently smooth functions.

For a smooth solution with bounded fourth derivatives on the unit square, the five-point local residual is $O(h^2)$. The smallest eigenvalue of the unscaled Dirichlet matrix is $8{\sin }^2(\pi h/2)\ge 8h^2$. With $M^2$ interior points and $M\asymp h^{-1}$, the residual has unweighted Euclidean norm $O(h)$. The equation $Ae=h^2\tau$ therefore gives unweighted error $O(h)$, equivalent to second-order error in the area-weighted grid norm.

On a unit-square uniform mesh with $h=1/(M+1)$ and zero Dirichlet boundary conditions, the semidiscrete five-point Laplacian has eigenvalues $-4h^{-2}[{\sin }^2(j\pi h/2)+{\sin }^2(k\pi h/2)]$. Adding reaction $\kappa I$ gives a contractive continuous-time grid evolution exactly under the displayed threshold. The threshold is $2{\pi }^2-{\pi }^4{h}^2/6+O(h^4)$, slightly below the continuum threshold. This is an all-time non-growth criterion; finite-time mesh-uniform bounds also hold for fixed larger reaction rates.