Subgrid Upscaling and Mixed Multiscale Finite Elements

Abstract

Second order elliptic problems in divergence form with a highly varying leading order coefficient on the scale epsilon can be approximated on coarse meshes of spacing H \gg epsilon only if one uses special techniques. The mixed variational multiscale method, also called subgrid upscaling, can be used, and this method is extended to allow oversampling of the local subgrid problems. The method is shown to be equivalent to the multiscale finite element method when one uses the lowest order Raviart--Thomas spaces and provided that there are no fine scale components in the source function f. In the periodic setting, a multiscale error analysis based on homogenization theory of the more general subgrid upscaling method shows that the error is O(epsilon+Hm + \sqrt epsilon/H ), where m=1. Moreover, m=2 if one uses the second order Brezzi-Douglas-Marini or Brezzi-Douglas-Durán-Fortin spaces and no oversampling. The error bounding constant depends only on the Hm - 1 -norm of f and so is independent of small scales when m=1. When oversampling is not used, a superconvergence result for the pressure approximation is shown.