Robust Localization of the Best Error with Finite Elements in the Reaction-Diffusion Norm

We consider the approximation in the reaction-diffusion norm with continuous finite elements and prove that the best error is equivalent to a sum of the local best errors on pairs of elements. The equivalence constants do not depend on the ratio of diffusion to reaction. We illustrate the usefulness of this result with two applications. First, we discuss robustness and locking properties of continuous finite elements with respect to the reaction-diffusion norm. Second, we derive local error functionals that ensure robust performance of adaptive tree approximation in the reaction-diffusion norm.