A manufactured smooth periodic density for the PIC grid-force estimate

Since $0<a<1$, the density $(1+a\cos y)\mathrm{/}(2\pi )$ is nonnegative and integrates to one. Convolving its constant and first Fourier modes with [F1] leaves the constant coefficient $1\mathrm{/}(2\pi )$ and multiplies the cosine coefficient by ${\alpha }_{\epsilon }$. Subtracting the neutralizing background gives the displayed source. Since $-\frac{d^2}{d{x}^2}\cos x=\cos x$, the unique zero-mean potential is ${\varphi }_{\epsilon }=(a{\alpha }_{\epsilon }\mathrm{/}(2\pi ))\cos x$, and differentiation yields the displayed $E_{\epsilon }$. A second convolution multiplies its sine mode by ${\alpha }_{\epsilon }$, giving ${\eta }_{\epsilon }\ast E_{\epsilon }=a{\alpha }_{\epsilon }^2(2\pi {)}^{-1}\sin x$.

Apply [F2] to this smooth source on an odd $N$ grid. Since $d=1$, the power $h^{2-d\mathrm{/}2}$ equals $h^{3\mathrm{/}2}$, and $L^{d\mathrm{/}2}=(2\pi {)}^{1\mathrm{/}2}$, giving the displayed estimate. The upper bound ${\gamma }_{N,2\pi }\le \pi \mathrm{/}2$ is uniform in $N$, and all fixed-$\epsilon$ derivative constants are finite by [F1]; both powers $h^{3\mathrm{/}2}$ and $h^2$ tend to zero. No limit as $\epsilon \to 0$ is taken.