 Explicit one-dimensional dual residual bound in L2

For $-(p{u}^{\prime }{)}^{\prime }+qu=f$ 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 $p_{\ast }=\min p>0$, $q_{\ast }=\min q>0$, $q^{\ast }=\max q$, one reliable constant isThe proof combines a dual $H^2$ estimate with two interval Poincaré inequalities. Constants are independent of the nonuniform mesh.