Past exam of the approximation theory and finite element methods course of the University of Oxford / 2008 / B / 5 / c / Solution

Testing against compactly supported smooth functions gives the distributional identity $u^{\prime \prime }=\cos xu-f$. The right side belongs to $L^2$, since $u\in H^1$ and $\cos x$ is bounded. Therefore $u^{\prime }\in H^1$ and $u\in H^2(0,1)$. This also makes the endpoint derivative traces meaningful, and integrating the weak form by parts recovers the Robin conditions.

The standard piecewise linear finite element interpolation estimate iswith constants independent of $h$. Hence $\Vert u-I_{h}u{\Vert }_{H^1}\le Ch\Vert u^{\prime \prime }{\Vert }_{L^2}$ for $h\le 1$. By Céa's lemma, with the fixed continuity and coercivity constants of $a$,The constant depends on the continuous problem but not on mesh refinement.