 
Summary: ANALYSIS OF A TWOSCALE, LOCALLY CONSERVATIVE
SUBGRID UPSCALING FOR ELLIPTIC PROBLEMS
TODD ARBOGAST
SIAM J. NUMER. ANAL. c 2004 Society for Industrial and Applied Mathematics
Vol. 42, No. 2, pp. 576598
Abstract. We present a twoscale 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 × V L2 × H(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 × V into coarse and "subgrid" subspaces
Wc × Vc and W × V such that (1) · Vc = Wc and · V = W, and (2) the space 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 welldefined operator taking Wc × Vc to
W × V, which is localized in space, and it is used to upscale, that is, to remove the subgrid from
the coarsescale problem. Using standard mixed finite element spaces, twoscale mixed spaces are
defined. A mixed approximation is defined, which can be viewed as a type of variational multiscale
method or a residualfree bubble technique. A numerical Green's function approach is used to make
the approximation to the subgrid operator efficient to compute. A mixed method operator is
defined for the twoscale approximation spaces and used to show optimal order error estimates.
