|
| |
T. Arbogast
UT Austin
A TWO-SCALE THEORETICAL AND NUMERICAL FRAMEWORK FOR APPROXIMATING THE SOLUTION TO AN ELLIPTIC EQUATION
We present a two-scale theoretical framework for approximating the
solution of a second order elliptic problem. Further numerical
approximation by a subgrid upscaling technique gives a computable
algorithm. The elliptic coefficient is assumed to vary on a scale
that is resolvable on a fine numerical grid. However, it is also
assumed that limits on computational power require that the
computations be performed on a coarse grid. We explicitly decompose
the differential problem into an coarse-scale operator coupled to a
fine-scale operator localized in space. Homogenization of the
coarse-scale operator provides an explicit two-scale system that can
be approximated on a coarse-grid. This system remains coupled to the
fine-scale operator, which is localized in space to a coarse-grid
element. We approximate it on a fine grid. An influence function
(numerical Greens function) technique allows us to solve these
subgrid-scale problems independently of the coarse-grid approximation.
The coarse-grid problem is modified to take into account the
subgrid-scale solution and solved as a large linear system of
equations posed over a coarse grid. Finally, the coarse scale
solution is corrected on the subgrid-scale, providing a fine-grid
scale representation of the original solution. Numerical examples
representing single-phase flow in a porous medium are presented. For
such examples, the elliptic coefficient is the rock permeability, and
it fits the situation considered.
|
