A Priori Feedback Estimates for Multiscale Reaction-Diffusion Systems. (12th March 2018)

• Record Type: Journal Article
• Authors: Lind, Martin; Muntean, Adrian
• Is Part Of: Numerical functional analysis and optimization. Volume 39:Number 4(2018)
• Page Start: 413
• Page End: 437
• Publication Date: 2018-03-12
• Subjects: Feedback finite element method -- Galerkin approximation -- micro–macro coupling -- multiscale reaction–diffusion systems; 35K57 -- 65M60 -- 35B27; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2017.1369996
• Languages: English
• ISSNs: 0163-0563

Abstract

ABSTRACT: 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 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 a priori feedback estimates is in principle applicable to a large class of systems of PDEs with dual structure admitting strong solutions.