多尺度反应扩散系统的先验反馈估计
A priori feedback estimates for multiscale reaction-diffusion systems
摘要：
研究了多尺度反应扩散系统在宏观和微观空间尺度上的逼近问题。尺度之间的耦合是通过微观-宏观通量条件完成的。我们的目标系统有一个典型的结构，在介质中的反应-扩散-流动问题的分布式微结构（也称为，双重孔隙材料）。除了确保基本估计的收敛两尺度半离散Galerkin近似，我们提供了一组{ \em先验}反馈估计和本地反馈误差估计，有助于设计一个分布式的高误差策略，允许计算效率的放大和缩小微观结构。反馈估计的差错控制依赖于两尺度能量、规律性和插值估计，以及对负责传播（多尺度）近似误差的源的精细簿记。基于{ \em a priori }反馈估计的工作技术原则上适用于一大类具有允许强解的对偶结构的偏微分方程系统。
英文摘要：
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.
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