Numerical averaging of non-divergence structure elliptic operators

Many important equations in science and engineering contain rapidly varying operators that cannot be practically Suffciently resolved for accurate solutions. In some cases it is possible to obtain approximate solutions by replacing the rapidly varying operator with an appropriately averaged operator. In this paper we use formal asymptotic techniques to recover a formula for the averaged form of a second order, non-divergence structure, linear elliptic operator. For several special cases the averaged operator is obtained analytically. For genuinely multi-dimensional cases, the averaged operator is also obtained numerically using a finite difference method, which also has a probabilistic interpretation.