A priori feedback estimates for multiscale reaction-diffusion systems

We study the approximation of a multiscale reaction-diffusion system posed on both macroscopic and microscopic space scales. The coupling between the scales is done via micro-macro flux conditions. Our target system has a typical structure for reaction-diffusion-flow problems in media with distributed microstructures (also called, double porosity materials). Besides ensuring basic estimates for the convergence of two-scale semi-discrete Galerkin approximations, we provide a set of {\em a priori} feedback estimates and a local feedback error estimator that help in designing a distributed-high-errors strategy to allow for a computationally efficient zooming in and out from microscopic structures. The error control on the feedback estimates relies on two-scale-energy, regularity, and interpolation estimates as well as on a fine bookeeping of the sources responsible with the propagation of the (multiscale) approximation errors. The working technique based on {\em a priori } feedback estimates is in principle applicable to a large class of systems of PDEs with dual structure admitting strong solutions.