Subgrid Upscaling and Mixed Multiscale Finite Elements

Second order elliptic problems in divergence form with a highly varying leading order coefficient on the scale can be approximated on coarse meshes of spacing H 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(+H m + /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 H m−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. 1. Introduction. Many physical problems can be modeled by a second order elliptic partial differential equation in space. In many cases, the coefficients of the equation are highly heterogeneous, which induces fine scale variability in the solution. Thus the difficulty in approximating the solution on a coarse finite element mesh T H is that the solution is not fully resolved on this scale. Traditional finite element analysis fails, and we require some multiscale approximation techniques. Babuška and Osborn [10, 9] proposed using special finite elements to approximate the solution. Hughes et al. [23, 24] (see also [13]) developed a more formal framework, which they called the variational multiscale method. A mixed variant, described as subgrid upscaling, was developed by Arbogast et al. [22] took a more direct approach and simply proposed finding a special finite element basis by solving the problem locally. They called this approach the multiscale finite element method. A mixed form was developed later by Chen and Hou [17]. To be more precise, consider a connected, convex polygonal domain Ω ⊆ R d , where d = 2 or 3, and a second order, uniformly positive-definite symmetric tensor a, so that both a and a −1 are uniformly elliptic and uniformly bounded. Suppose