Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 38A (Numerical Analysis)

Assume sufficient smoothness up to the boundary for uniform Taylor remainders, for example $u\in C^8(\overline{\Omega })$ and $f=\Delta u\in C^6(\overline{\Omega })$. Direct Taylor expansion of the nine-point finite-difference stencil givesThus the normalized local truncation error $h^{-2}{\Gamma }_{9}u-f$ is $\boxed{O(h^2)}$. The unscaled residual ${\Gamma }_{9}u-h^{2}f$ is $O(h^4)$. Stating the normalization avoids ambiguity in the meaning of truncation order.

If $\Delta f=0$, then ${\Delta }^{2}u=0$, so the $h^4$ term in the previous expansion vanishes. Therefore the normalized local truncation error is $\boxed{O(h^4)}$, and the unscaled residual is $O(h^6)$. This is generally fourth order under the given harmonic-source condition; special solutions can have additional cancellation, but that is not guaranteed by this argument.

The five-point Laplacian applied to $f$ has expansionThe corrected right side $h^{2}f+(h^2/12){\Gamma }_{5}f$ cancels the $h^4\Delta f/12$ term in ${\Gamma }_{9}u$. ConsequentlyMore precisely, the unscaled residual's $h^6$ coefficient is $-(u_{xxxxxx}+u_{yyyyyy})/240+(u_{xxxxyy}+u_{xxyyyy})/144$. This fourth-order correction of the nine-point Poisson stencil does not require $f$ to be harmonic.

Let $A$ be the interior-grid matrix representing ${\Gamma }_9$, with zero boundary error. It is real symmetric and has the stated sine eigenbasis. Put $s={\sin }^2(k\pi h/2)$ and $t={\sin }^2(\ell \pi h/2)$; the negative of its eigenvalue isThis increases in each variable on $[0,1{]}^2$. With $s_1={\sin }^2(\pi h/2)\ge h^2$, the smallest absolute eigenvalue is at least $8s_1-(8/3)s_1^2\ge (16/3)h^2$. Hence its Euclidean norm operator norm satisfiesThe corrected scheme has an unscaled residual $r$ whose individual entries are uniformly $O(h^6)$. There are $m^2$ entries and $m<h^{-1}$, so $\Vert r{\Vert }_2=O(h^5)$. Subtracting the exact nodal equations from the numerical equations gives $Ae=-r$ (with sign depending on the residual convention), and thereforeChoose a constant slightly larger than the resulting bound to obtain the requested strict $\Vert e\Vert <c{h}^3$ for all sufficiently small $h$. Uniform smoothness is needed for an $h$-independent constant; unspecified smoothness only in the interior would not by itself guarantee it.