P1 edge interpolation estimate with junction compatibility

Fix an edge cell $K=[a,b]$ of length $h_K=b-a$. By [F1] and [F2], the continuous representative of $u_e$ has endpoint values and the representative of $u_e^{\mathrm{\prime }}$ is absolutely continuous. Thus the slope of the nodal interpolant on $K$ is $s_K=\frac{u_e(b)-u_e(a)}{h_K}=\frac{1}{h_K}{\int }_a^b{u}_e^{\mathrm{\prime }}(y)dy\mathrm{.}$ The fundamental theorem in [F2] justifies this equality.

For any $x\in K$, [F2] and [F3] give $\mathrm{\mid }{u}_e^{\mathrm{\prime }}(x)-s_K\mathrm{\mid }\le \frac{1}{h_K}{\int }_a^b\mathrm{\mid }{u}_e^{\mathrm{\prime }}(x)-u_e^{\mathrm{\prime }}(y)\mathrm{\mid }dy\le \frac{1}{h_K}{\int }_a^b\sqrt{h_K}\mathrm{\parallel }{u}_e^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{L^2(K)}dy=\sqrt{h_K}\mathrm{\parallel }{u}_e^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{L^2(K)}\mathrm{.}$ Squaring and integrating over $x\in K$ yields $\mathrm{\parallel }(u_e-I_h{u}_e{)}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2(K)}^2=\mathrm{\parallel }{u}_e^{\mathrm{\prime }}-s_K{\mathrm{\parallel }}_{L^2(K)}^2\le h_K^2\mathrm{\parallel }{u}_e^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{L^2(K)}^2\mathrm{.}$ The first equality holds because the derivative of the nodal interpolant is the constant slope $s_K$ on K.

Multiply the cell estimate in step 2.1 by ${\kappa }_e$, sum over all cells, and use $h_K\le h$. The cells partition the edge interiors and their finitely many endpoints have measure zero. Hence ${\sum }_e{\kappa }_e\mathrm{\parallel }(u_e-I_h{u}_e{)}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2(e)}^2\le h^2{\sum }_e{\kappa }_e\mathrm{\parallel }{u}_e^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{L^2(e)}^2\mathrm{.}$ Taking square roots proves the displayed error estimate.

At an original vertex, the endpoint nodal value at every incident endpoint incidence is the common trace of $u$, so all interpolant traces agree, including both ends of a loop. At each anchor the nodal value is zero. The interpolant is real-valued, continuous and piecewise affine on each finite edge partition, so its weak derivative is its piecewise constant slope in $L^2$ (the internal endpoint terms cancel when testing against a smooth compactly supported function). Hence it belongs to the edge-product real $H^1$ space. Thus $I_{h}u\in V_D$ and it is affine on every cell by construction, which is exactly membership in $V_h$.