I am reading a paper that there a question confuses me. It says that ''...use $r+1$-point Gauss-Lobatto quadrature on the interval $I = [a,b](b-a = h)$, Since the Gauss-Lobatto quadrature on $[a,b]$ is exact for polynomials of degree $2r-1$, employing the Bramble-Hilbert lemma can derive : $$ |(\Delta u,\phi)_h - (\Delta u, \phi)| = | \sum_{j=0}^{r} w_j (\Delta u \phi)(x_{(i-1)r+j}) - \int_{I} (\Delta u) \phi dx) | $$ $$ \leq Ch^{2r} \| \Delta u \phi\|_{W^{2r,1}\;(I)} \quad \leq Ch^{2r} \|u\|_{H^{2r+2}\;(I)} \|\phi\|_{H^{r}\;(I)}. $$ where $w_j$ is the weight of intergration points, and $W^{k,p}$ is the Sobolev space.
The norm $\|u\|_{H^{2r+2}\;}$ means that $u$ is required to have high smoothness, and I want to know how the first inequality is derived and what is the Bramble-Hilbert lemma? Thx！