= Explicit one-dimensional dual residual bound in L2
{title2=$\|u-u_h\|_2\leq C(\sum_i h_i^4\|R_h\|_{2,I_i}^2)^{1/2}$}
For $-(pu')'+qu=f$ with one homogeneous Neumann condition and one fixed Dirichlet value, the dual problem has error forcing and homogeneous corresponding boundary data. <Galerkin orthogonality> with its nodal interpolant yields an element-residual identity with no jump terms, because the interpolation error vanishes at every node. With $p_*=\min p>0$, $q_*=\min q>0$, $q^*=\max q$, one reliable constant is
$$
C=\frac1{\pi^2p_*}\left(1+\frac{q^*}{q_*}
+\frac{\|p'\|_\infty}{\sqrt{p_*q_*}}\right).
$$
The proof combines a dual $H^2$ estimate with two interval Poincaré inequalities. Constants are independent of the nonuniform mesh.
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