Matching variable-degree polynomial traces give an H1 function

If the mesh is empty, its domain is empty by the mesh convention and the assertion is immediate. Otherwise, because the mesh and every local degree are finite, $v$ and each cellwise first derivative $g_i:={\mathrm{\partial }}_i(v{\mathrm{\mid }}_K)$ are bounded on each closed cell; a finite union then puts $v,g_i$ in $L^2(\mathrm{\Omega })$. If the mesh is simplicial, set $p_{\ast }={\max }_{K\in \mathcal{T}}{p}_K$. Every restriction lies in ${\mathbb{P}}_{p_{\ast }}(K)$, and the trace-matching condition is unchanged. Applying [F2] at the single degree $p_{\ast }$ proves the claim on this branch.

Suppose the cells are affine tensor boxes. Fix a coordinate $i$. On each cell, apply [F3] to the smooth field $v\phi e_i$; its divergence is $v{\mathrm{\partial }}_i\phi +g_i\phi$. Sum the finitely many resulting identities. At an interior face the two outward normals are opposite, because the adjacent cell interiors lie on opposite sides of its supporting hyperplane; the common trace in [F1] therefore makes the two fluxes cancel. The boundary-face terms vanish since $\phi$ has compact support in $\mathrm{\Omega }$. Thus ${\int }_{\mathrm{\Omega }}v{\mathrm{\partial }}_i\phi =-{\int }_{\mathrm{\Omega }}{g}_i\phi$.

By [F4], step 1.2 says that $g_i$ is the weak $i$th derivative of $v$. This holds for every coordinate $i$, and step 1.1 supplies $v,g_i\in L^2$; hence $v\in H^1(\mathrm{\Omega })$ with the stated cellwise weak gradient. Together with the simplex branch of step 1.1 this proves the assertion for both mesh types.