= Harmonic superconvergence of the nine-point stencil
{title2=$\Delta u=0\Longrightarrow D_9u=O(h^6)$}
For the square-grid <nine-point finite-difference stencil> for the <Laplacian>,
$$
D_9u=\Delta u+\frac{h^2}{12}\Delta^2u+\frac{h^4}{360}(u_{xxxxxx}+u_{yyyyyy}+5u_{xxxxyy}+5u_{xxyyyy})+O(h^6).
$$
If $\Delta u=0$, both displayed correction terms vanish: the mixed sixth <derivatives> sum to $\Delta u_{xxyy}=0$, and the pure sixth <derivatives> sum to the negative of that sum. Expanding two orders further yields $D_9u=h^6u_{xxxxxxxx}/3024+O(h^8)$ for a smooth <harmonic function>. Thus the compact harmonic scheme has sixth-order normalized <consistency of a numerical method> and, with the <discrete maximum principle> inverse bound, sixth-order nodal convergence on a square with exact smooth <Dirichlet boundary data>. For example $\operatorname{Re}(x+iy)^8$ shows that the sixth-order term need not vanish.
Back to article page
