Second-order max-norm error for the centered Dirichlet scheme

Let ${\tau }_h=A_h{u}_{\mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d}}-g_h$. By [F3], $\mathrm{\parallel }{\tau }_h{\mathrm{\parallel }}_{\mathrm{\infty }}\le M{h}^2\mathrm{/}12$ and the exact endpoint values agree with the imposed endpoint closure, so the endpoint error is zero.

For $e_h:=v_h-u_{\mathrm{g}\mathrm{r}\mathrm{i}\mathrm{d}}$, subtracting the definitions of ${\rho }_h$ and ${\tau }_h$ gives $A_h{e}_h={\rho }_h-{\tau }_h$. Since $A_h$ is invertible by [F2], $e_h=A_h^{-1}({\rho }_h-{\tau }_h)$.

By [F2], each coordinate satisfies $\mathrm{\mid }(e_h{)}_j\mathrm{\mid }\le \mathrm{\parallel }{\rho }_h-{\tau }_h{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{/}8$. The coordinatewise triangle inequality [F4] gives $\mathrm{\parallel }{\rho }_h-{\tau }_h{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }{\rho }_h{\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\parallel }{\tau }_h{\mathrm{\parallel }}_{\mathrm{\infty }}$, because for each $j$, $\mathrm{\mid }{\rho }_j-{\tau }_j\mathrm{\mid }\le \mathrm{\parallel }{\rho }_h{\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\parallel }{\tau }_h{\mathrm{\parallel }}_{\mathrm{\infty }}$.

Substituting the residual bound from [F3] into step 3.1 yields $\mathrm{\parallel }{e}_h{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }{\rho }_h{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{/}8+M{h}^2\mathrm{/}(8\cdot 12)=\mathrm{\parallel }{\rho }_h{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{/}8+M{h}^2\mathrm{/}96$. For the exact algorithm output, [F2] guarantees that the recurrence returns the unique solution and the exact equation check succeeds, so ${\rho }_h=0$; exact endpoint closure was recorded in step 1.1.