We study the homogenization of the equation $-A(\frac{\cdot}{\varepsilon}):D^2 u_{\varepsilon} = f$ posed in a bounded convex domain $\Omega\subset \mathbb{R}^n$ subject to a Dirichlet boundary condition and the numerical approximation of the corresponding homogenized problem, where the measurable, uniformly elliptic, periodic and symmetric diffusion 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 and uniqueness of an invariant measure by reducing to a Lax--Milgram-type problem, we obtain $L^2$-bounds for periodic problems in double-divergence-form, we prove homogenization under minimal regularity assumptions, and we generalize known corrector bounds and results on optimal convergence rates from the classical case of H\"{o}lder continuous coefficients to the present case. In the second part, we suggest and rigorously analyze an approximation scheme for the effective coefficient matrix and the solution to the homogenized problem based on a finite element method for the approximation of the invariant measure, and we demonstrate the performance of the scheme through numerical experiments.

翻译：我们研究方程 $-A(\frac{\cdot}{\varepsilon}):D^2 u_{\varepsilon} = f$ 在有界凸域 $\Omega\subset \mathbb{R}^n$ 中受狄利克雷边界条件的齐次化问题，以及相应齐次化问题的数值逼近。其中，可测、一致椭圆、周期且对称的扩散矩阵 $A$ 仅假设为本质有界且（若 $n>2$）满足Cordes条件。第一部分中，我们通过归约为Lax-Milgram型问题证明了不变测度的存在唯一性，获得了双散度型周期问题的$L^2$界，在最小正则性假设下证明了齐次化，并将经典Hölder连续系数情形下的已知校正子界和最优收敛率结果推广至当前情形。第二部分中，我们提出并严格分析了基于不变测度有限元逼近的有效系数矩阵和齐次化问题解的逼近格式，并通过数值实验展示了该格式的性能。