L2-GLOBAL TO LOCAL PROJECTION: AN APPROACH TO MULTISCALE ANALYSIS

Abstract

The paper introduces a new global–local method for recovering the microscale features of gradient fields inside heterogeneous media. The approach is based upon the L2-projection of the homogenized solution onto function spaces spanned by solutions of local problems. The projection framework introduced is general and applies to local domains ω on the interior of the domain of interest as well as those touching the boundary. Within this new framework it is shown that the difference between the actual solution and the L2-projection of the homogenized solution as measured in the energy norm over ω is the same order as the error between the homogenized solution and the actual solution as measured by the L2-norm over a slightly larger domain. In light of the classic results from periodic homogenization this is the best one can do see A. Besounssan, J. L. Lions and G. C. Papanicolau, Asymptotic Analysis for Periodic Structures (North-Holland, 1978).