 Solution

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.