Let us consider $\Omega\subset\mathbb{R}^d$, $d=2,3$ polyhedral domain, i.e. $\bar{\Omega}$ is the union of a finite number of polyhedra.
Let $\bar{\Omega}=\bigcup_{K\in\mathcal{T}_h}K$, where $\mathcal{T}_h$ is a triangulation of simplexes.
I want to prove the following result.
A function $v:\Omega\to\mathbb{R}$ belongs to $H^1\left(\Omega\right)$ if and only if
- $v|_{K}\in H^1\left(K\right)$ for each $K\in\mathcal{T}_h$,
- $v|_{K}\in H^1\left(K\right)$ for each $K\in\mathcal{T}_h$, for each common face $F=K_1\cap K_2$, $K_1,K_2\in\mathcal{T}_h$, the trace of $v|_{K_1}$ and $v|_{K_2}$ on $F$ is the same.
I proved the reverse implication "$\leftarrow$", but I have troubles with the other one.
Assume $v\in H^1\left(K\right)$. We have, of course, that $v|_{K}\in H^1\left( K\right)$ for each $K\in\mathcal{T}_h$, which implies that the trace on $F$ is well defined. We need to prove that the last statement about the traces holds.
Any ideas?
I know that I should end up with $$ \sum_F\int_F\left({v}|_{K_1}-{v}|_{K_2} \right)\varphi n_j=0 $$ for every $\varphi\in C^{\infty}_0(\Omega)$.