 Solution

Put $e=u-u_h$ and $V=\{v\in H^1(0,1):v(1)=0\}$. The primal solution and its finite element approximation lie in the same affine space with value one at the right endpoint, so $e\in V$. UseDefine the dual solution byEquivalently $z\in V$ satisfies $a(v,z)=(e,v)$ for every $v\in V$. Positivity of $p,q$ gives a unique solution, and one-dimensional regularity gives $z\in H^2$.

Then $\Vert e{\Vert }_{L^2}^2=a(e,z)$. The nodal interpolant $I_{h}z$ belongs to the homogeneous finite element test space, so Galerkin orthogonality gives $a(e,I_{h}z)=0$. Set $w=z-I_{h}z$. Using the primal weak equation,Integrate the derivative term separately on each element. Since $w$ vanishes at every mesh node, all endpoint flux and flux-jump terms vanish. ThereforeThe absence of jump terms here follows specifically from one-dimensional nodal interpolation; it is not an assumption that the finite element flux itself is continuous.