Anchored finite metric graph and its H1 energy space

Assume the Axiom of Choice as required by the cited one-dimensional representative interface. A finite metric graph $\mathrm{\Gamma }=(V,E,{\mathrm{\partial }}_{-},{\mathrm{\partial }}_{+},\mathrm{\ell })$ consists of finite vertex and edge sets $V,E$, endpoint maps ${\mathrm{\partial }}_{-},{\mathrm{\partial }}_{+}:E\to V$, and edge lengths $0<{\mathrm{\ell }}_e<\mathrm{\infty }$. Distinct edges with the same endpoints are allowed, as are loop edges with ${\mathrm{\partial }}_{-}e={\mathrm{\partial }}_{+}e$. Every vertex is an endpoint of at least one edge. The two endpoint incidences $(e,-)$ and $(e,+)$ remain distinct even for a loop. A graph path is a finite sequence $z_0,e_1,z_1,\dots ,e_n,z_n$ with $n\ge 0$ and $\{{\mathrm{\partial }}_{-}{e}_j,{\mathrm{\partial }}_{+}{e}_j\}=\{z_{j-1},z_j\}$ for every $j$; a loop permits $z_{j-1}=z_j$. Vertices in the same connected edge component are those joined by such a path. Orient each edge only to choose its coordinate $x_e\in [0,{\mathrm{\ell }}_e]$, with $0$ at ${\mathrm{\partial }}_{-}e$ and ${\mathrm{\ell }}_e$ at ${\mathrm{\partial }}_{+}e$. For $z\in V$, let $\mathcal{I}(z):=\{(e,-):{\mathrm{\partial }}_{-}e=z\}\cup \{(e,+):{\mathrm{\partial }}_{+}e=z\}$ be its finite set of endpoint incidences. The edge-product spaces are

$H:={⨁}_{e\in E}{L}^2(0,{\mathrm{\ell }}_e;\mathbb{R}),\mathcal{W}:={⨁}_{e\in E}{H}^1(0,{\mathrm{\ell }}_e;\mathbb{R})\mathrm{.}$

For $w_e\in H^1(0,{\mathrm{\ell }}_e)$, its endpoint values mean the values of the unique absolutely continuous representative supplied by One-dimensional $W^{1,p}$ functions have unique absolutely continuous representatives. The trace-matching graph Sobolev space is the subspace of $\mathcal{W}$ whose endpoint-incidence traces all agree at each vertex. For a designated anchor set $D\subseteq V$, define the anchored graph energy space

$V_D:=\{w\in \mathcal{W}:w_{\iota }=w_{{\iota }^{\mathrm{\prime }}}\text{ whenever }\iota ,{\iota }^{\mathrm{\prime }}\in \mathcal{I}(v),w_{\iota }=0\text{ for each }v\in D,\iota \in \mathcal{I}(v)\}\mathrm{.}$

We equip $H$ with the edge-sum $L^2$ inner product and $V_D$ with

$\mathrm{\parallel }w{\mathrm{\parallel }}_{V_D}^2={\sum }_{e\in E}{\int }_0^{{\mathrm{\ell }}_e}(\mathrm{\mid }{w}_e{\mathrm{\mid }}^2+\mathrm{\mid }{w}_e^{\mathrm{\prime }}{\mathrm{\mid }}^2)d{x}_e\mathrm{.}$

No Kirchhoff derivative condition is built into $V_D$; that condition is a natural boundary condition of the later weak diffusion form. The elliptic and P1 error results below assume that $D$ meets every connected edge component.