Among the second order elliptic equations that arise frequently in science and engineering are the Monge-Ampère equation and the non-divergence structure linear equation with rapidly oscillating coefficients. Both of these PDEs are challenging from a numerical viewpoint: the first due to its nonlinearity and the second due to the impracticality of properly resolving the coefficients. In the first part of this thesis we construct finite difference schemes for the two-dimensional Monge-Ampère equation. Numerical investigation indicates that these schemes converge even in situations where smooth solutions do not exist. Secondly, we use formal asymptotic techniques to recover an appropriate average of the non-divergence structure operator. For several special cases we construct this operator in closed form. When this is not possible, the averaged operator is obtained numerically. Numerical investigations are consistent with the assertion that solutions of the original equation converge to solutions of the averaged equation.
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