This contribution is the numerically oriented companion article of the work [9]. We focus here on the numerical resolution of the embedded corrector problem introduced in [8], [9] in the context of homogenization of diffusion equations. Our approach consists in considering a corrector-type problem, posed on the whole space, but with a diffusion matrix which is constant outside some bounded domain. In [9], we have shown how to define three approximate homogenized diffusion coefficients on the basis of the embedded corrector problem. We have also proved that these approximations all converge to the exact homogenized coefficients when the size of the bounded domain increases.

We show here that, under the assumption that the diffusion matrix is piecewise constant, the corrector problem to solve can be recast as an integral equation. In case of spherical inclusions with isotropic materials, we explain how to efficiently discretize this integral equation using spherical harmonics, and how to use the fast multipole method (FMM) to compute the resulting matrix-vector products at a cost which scales only linearly with respect to the number of inclusions. Numerical tests illustrate the performance of our approach in various settings.

该贡献是该工作的数字导向伴侣文章[9]。在这里，我们集中讨论在扩散方程的均化情况下，在[8]，[9]中引入的嵌入校正器问题的数值分辨率。我们的方法包括考虑一个校正器类型的问题，该问题摆在整个空间上，但扩散矩阵在某些有界域之外是恒定的。在[9]中，我们展示了如何根据嵌入的校正器问题定义三个近似的均匀扩散系数。我们还证明，当有界域的大小增加时，这些近似值都收敛到精确的均化系数。

我们在这里表明，在假设扩散矩阵为分段常数的前提下，可以将要解决的校正器问题重新积分为一个积分方程。对于各向同性材料的球形夹杂物，我们将说明如何使用球谐函数有效地离散该积分方程，以及如何使用快速多极方法（FMM）计算生成的矩阵矢量乘积，其成本仅相对于线性夹杂物的数量。数值测试说明了我们的方法在各种环境下的性能。