I'm trying to enderstand the equations below for an estimate of the a-priori error in the energy norm. My professor writes in his notes that
\begin{align} ||e||_E^2 &= \int_0^1e'(u-v)'+2xe'(u-v)+2e(u-v) \ dx \\ &\leq 2||e'||\cdot||u-v||_E+2||e||\cdot ||u-v|| \\ &\leq 2||e||_E\cdot ||u-v||_E \end{align}
and derive the a priori error estimate
$$||e||_E\leq (1+\alpha)||u-v||_E , \quad \forall v\in V_h.$$
Questions:
1) What happens to the $x$ in the integrand?
2) Why do only some of the norms get an $E$ as an index and not all of them?
3) In the second $\rightarrow$ third step, he just removes the $2||e'||\cdot ||u-v||_E$ I assume? If that's the case, then we should only have been left with $2||e||\cdot |||u-v||$, why indexing the norms with $E$'s? I'm starting to think that $||.||=||.||_E$ and that he just forgot to add them to all.
4) Where does $\alpha$ come from? Says nothing anywhere about the nature of $\alpha$ for this problem.