A Priori Feedback Estimates for Multiscale Reaction-Diffusion Systems

作者：Lind Martin; Muntean Adrian

来源：Numerical Functional Analysis and Optimization, 2018, 39(4): 413-437.

DOI：10.1080/01630563.2017.1369996

• Feedback finite element method
• Galerkin approximation
• micro-macro coupling
• 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 ecient 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.

• 出版日期2018