Finite Element Approximation of Elliptic Homogenization Problems in
Nondivergence-Form

We use uniform $W^{2,p}$ estimates to obtain corrector results for periodic
homogenization problems of the form $A(x/\varepsilon):D^2 u_{\varepsilon} = f$
subject to a homogeneous Dirichlet boundary condition. We propose and
rigorously analyze a numerical scheme based on finite element approximations
for such nondivergence-form homogenization problems. The second part of the
paper focuses on the approximation of the corrector and numerical
homogenization for the case of nonuniformly oscillating coefficients. Numerical
experiments demonstrate the performance of the scheme.