 
Summary: SIAM J. NUMER. ANAL. c 2006 Society for Industrial and Applied Mathematics
Vol. 44, No. 3, pp. 11501171
SUBGRID UPSCALING AND MIXED MULTISCALE FINITE
ELEMENTS
TODD ARBOGAST AND KIRSTEN J. BOYD
Abstract. 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
RaviartThomas 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( +Hm+ /H), where m = 1. Moreover,
m = 2 if one uses the second order BrezziDouglasMarini or BrezziDouglasDurŽanFortin spaces
and no oversampling. The error bounding constant depends only on the Hm1norm 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.
Key words. mixed method, multiscale finite element, subgrid upscaling, variational multiscale
AMS subject classifications. 65N15, 65N30, 35J20
DOI. 10.1137/050631811
