Exponential Estimates in Averaging and Homogenisation [chapter]

Karsten Matthies
Analysis, Modeling and Simulation of Multiscale Problems  
Many partial differential equations with rapid spatial or temporal scales have effective descriptions which can be derived by homogenisation or averaging. In this article we deal with examples, where quantitative estimates of the error is possible for higher order homogenisation and averaging. In particular, we provide theorems, which allow homogenisation and averaging beyond all orders by giving exponential estimates of appropriately averaged and homogenised descriptions. Methods include
more » ... thods include iterated averaging transformations, optimal truncation of asymptotic expansions and highly regular solutions (Gevrey regularity). Prototypical examples are reaction-diffusion equations with heterogeneous reaction terms or rapid external forcing, nonlinear Schrödinger equations describing dispersion management, and second-order linear elliptic equations.
doi:10.1007/3-540-35657-6_1 fatcat:p22p74kvungwtotfrtmijrtx6m