SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 646-666, April 2024.
Abstract. We study the homogenization of the equation [math] posed in a bounded convex domain [math] 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 [math] is merely assumed to be essentially bounded and (if [math]) 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 [math]-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 Hölder 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.

中文翻译：

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具有间断系数的非散度型椭圆方程的齐次化及其齐次问题的有限元逼近

SIAM 数值分析杂志，第 62 卷，第 2 期，第 646-666 页，2024 年 4 月。
摘要。我们研究在狄利克雷边界条件下的有界凸域 [math] 中提出的方程 [math] 的均质化以及相应均质化问题的数值近似，其中可测量的均匀椭圆、周期和对称扩散矩阵 [math] ] 只是假设本质上是有界的并且（如果 [math]）满足 Cordes 条件。在第一部分中，我们通过简化为 Lax-Milgram 型问题来证明不变测度的存在性和唯一性，我们以双散度形式获得周期问题的[数学]边界，我们证明了最小正则性假设下的同质化，我们将已知的校正器界限和最佳收敛率结果从霍尔德连续系数的经典情况推广到当前情况。在第二部分中，我们提出并严格分析了有效系数矩阵的近似方案以及基于有限元方法近似不变测度的均质问题的解，并通过数值实验证明了该方案的性能。