Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems

We present a two-scale theoretical framework for approximating the solution of a second order elliptic problem. The elliptic coefficient is assumed to vary on a scale that can be resolved on a. ne numerical grid, but limits on computational power require that computations be performed on a coarse grid. We consider the elliptic problem in mixed variational form over W x V subset of L-2 x H(div). We base our scale expansion on local mass conservation over the coarse grid. It is used to de. ne a direct sum decomposition of W x V into coarse and "subgrid" subspaces W-c x V-c and deltaW x deltaV such that (1) del. V-c = W-c and del . deltaV = deltaW, and (2) the space deltaV is locally supported over the coarse mesh. We then explicitly decompose the variational problem into coarse and subgrid scale problems. The subgrid problem gives a well-defined operator taking Wc x Vc to dW x dV, which is localized in space, and it is used to upscale, that is, to remove the subgrid from the coarse-scale problem. Using standard mixed finite element spaces, two-scale mixed spaces are defined. A mixed approximation is defined, which can be viewed as a type of variational multiscale method or a residual-free bubble technique. A numerical Green's function approach is used to make the approximation to the subgrid operator efficient to compute. A mixed method pi-operator is defined for the two-scale approximation spaces and used to show optimal order error estimates.