

Todd Arbogast: Twoscale, locally conservative
variational multiscale methods for elliptic problem
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 that limits on computational
power require that computations be performed on a coarse grid. We
consider the elliptic problem in mixed variational form, and base our
scale expansion on local mass conservation over a coarse grid. The
coarse grid is used to define a direct sum decomposition of the
solution space into coarse and ``subgrid'' subspaces satisfying the
usual divergence constraints and the property that the subgrid space
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
the coarse space to the subgrid 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, twoscale 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 Greens function approach is used to
make the approximation to the subgrid operator efficient to compute.
A mixed method $\pi$operator is defined for the twoscale
approximation spaces and used to show optimal order error estimates.
Moreover, both the coarse and finescale problems remain locally
conservative. Numerical examples representing flow in a porous medium
are presented to illustrate the effectiveness and applicability of the
method. We also consider the use of the approximation as a
preconditioner for a direct approximation of the full problem.
