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We study the homogenization of the equation$-A(\frac{\cdot}{\varepsilon}):D^2 u_{\varepsilon} = f$ posed in a boundedconvex domain $\Omega\subset \mathbb{R}^n$ subject to a Dirichlet boundarycondition and the numerical approximation of the corresponding homogenizedproblem, where the measurable, uniformly elliptic, periodic and symmetricdiffusion matrix $A$ is merely assumed to be essentially bounded and (if $n>2$)to satisfy the Cordes condition. In the first part, we show existence anduniqueness of an invariant measure by reducing to a Lax--Milgram-type problem,we obtain $L^2$-bounds for periodic problems in double-divergence-form, weprove homogenization under minimal regularity assumptions, and we generalizeknown corrector bounds and results on optimal convergence rates from theclassical case of H\"{o}lder continuous coefficients to the present case. Inthe second part, we suggest and rigorously analyze an approximation scheme forthe effective coefficient matrix and the solution to the homogenized problembased on a finite element method for the approximation of the invariantmeasure, and we demonstrate the performance of the scheme through numericalexperiments.

Homogenization of nondivergence-form elliptic equations with discontinuous coefficients and finite element approximation of the homogenized problem