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ANALYSIS OF A TWO-SCALE, LOCALLY CONSERVATIVE SUBGRID UPSCALING FOR ELLIPTIC PROBLEMS
 

Summary: ANALYSIS OF A TWO-SCALE, 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 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 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 well-defined 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 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 -operator is
defined for the two-scale approximation spaces and used to show optimal order error estimates.

  

Source: Arbogast, Todd - Center for Subsurface Modeling & Department of Mathematics, University of Texas at Austin

 

Collections: Mathematics; Geosciences