Cea and O(h) energy error for conforming metric graph P1 FEM

The map sending a P1 graph function to its finite list of nodal values is linear and injective, since an affine function on a cell is determined by its two endpoint values. There are finitely many cells, so $V_h$ is finite-dimensional. Its defining vertex and anchor constraints make it a subspace of $V_D$.

For $w,v\in V_D$, [F7] gives $\mathrm{\mid }a(w,v)\mathrm{\mid }\le {\kappa }_{\max }\mathrm{\parallel }{w}^{\mathrm{\prime }}{\mathrm{\parallel }}_H\mathrm{\parallel }{v}^{\mathrm{\prime }}{\mathrm{\parallel }}_H\le M_a\mathrm{\parallel }w{\mathrm{\parallel }}_{V_D}\mathrm{\parallel }v{\mathrm{\parallel }}_{V_D}$. By path Poincare, $\mathrm{\parallel }v{\mathrm{\parallel }}_{V_D}^2\le (1+L_{\mathrm{\Gamma }}^2)\mathrm{\parallel }{v}^{\mathrm{\prime }}{\mathrm{\parallel }}_H^2$, so $a(v,v)\ge {\kappa }_{\min }\mathrm{\parallel }{v}^{\mathrm{\prime }}{\mathrm{\parallel }}_H^2\ge {\alpha }_a\mathrm{\parallel }v{\mathrm{\parallel }}_{V_D}^2$. Thus $a$ is bounded and coercive with the stated constants and is symmetric.

Apply [F2] with $V=V_D$, subspace $V_h$, form $a$, and load $F$ from the weak problem. Its Hilbert, finite-dimensional inclusion, boundedness, coercivity and bounded-load hypotheses hold by [F1], step 1.1, and step 1.2. Hence a unique Galerkin solution exists and satisfies the displayed Cea estimate. It agrees with the restriction of the same weak equation because the continuous solution is unique by [F6].

Apply the symmetric best-approximation fact [F3] to $u_h$ and choose $v_h=I_{h}u$. By [F4], this interpolant is in $V_h$ and its energy error is at most $h({\sum }_e{\kappa }_e\mathrm{\parallel }{u}_e^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{L^2(e)}^2{)}^{1\mathrm{/}2}$. Therefore the Galerkin energy error is bounded by the same quantity, as claimed. The rate is conditional on the stated edgewise H2 regularity; no vertex regularity is inferred. By [F5], the natural free-vertex condition is Kirchhoff flux balance.