Chapter 6: Effective Coefficients and Periodic Homogenization
Axel Målqvist, Daniel Peterseim · Society for Industrial and Applied Mathematics eBooks · 2020
This chapter connects the LOD method and the mathematical theory of homogenization in the multidimensional setting. In one dimension it was possible to reconstruct the homogenized diffusion coefficient from the method. Following this idea, the bilinear form is rewritten as an integral operator acting on finite element spaces with an exponentially decaying discrete kernel that may be seen as a quasi-local effective representation of the rough diffusion coefficient on a coarser scale. The diagonal approximation of the kernel links the method to an effective PDE with an effective coefficient that is piecewise constant with respect to the coarse mesh H; this is similar to Section 2.3. In a periodic model problem this procedure reproduces the homogenized coefficient in the mathematical theory of homogenization.