Residual adaptation with a uniform-refinement safeguard

A reliable residual estimator certifies an error tolerance only when its upper bound is below the tolerance. To ensure eventual termination, residual-based element marking can be combined with occasional global bisection. For one-dimensional piecewise linear reaction-diffusion approximation, $R_h=f+p^{\prime }{u}_h^{\prime }-q{u}_h$ is bounded in $L^2$ by discrete energy stability, and the weighted estimator is at most $h_{\max }^2\Vert R_h{\Vert }_2$. Thus the safeguard $h_{\max }\to 0$ forces the estimator to zero and makes the certified stopping test eventually succeed.