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 through 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 a priori feedback estimates and a local feedback error estimator that help in designing a distributed-high-errors strategy to allow for a computationally e?cient 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 a priori feedback estimates is in principle applicable to a large class of systems of PDEs with dual structure admitting strong solutions.