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 fine 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 $W \times {\bf V} \subset L^2 \times H({\rm div})$. We base our scale expansion on local mass conservation over the coarse grid. It is used to define a direct sum decomposition of $W \times {\bf V}$ into coarse and "subgrid" subspaces $W_c \times {\bf V}_c$ and $\delta W \times \delta{\bf V}$ such that (1) $\nabla \cdot {\bf V}_c=W_c$ and $\nabla \cdot \delta{\bf V} = \delta W$, and (2) the space $\delta{\bf V}$ 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 $W_c \times {\bf V}_c$ to $\delta W \times \delta{\bf V}$, 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.