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This contribution is the numerically oriented companion article of the work[E. Canc\`es, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, arxiv preprint1807.05131]. We focus here on the numerical resolution of the embeddedcorrector problem introduced in [E. Canc\`es, V. Ehrlacher, F. Legoll and B.Stamm, CRAS 2015; E. Canc\`es, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang,arxiv preprint 1807.05131] in the context of homogenization of diffusionequations. Our approach consists in considering a corrector-type problem, posedon the whole space, but with a diffusion matrix which is constant outside somebounded domain. In [E. Canc\`es, V. Ehrlacher, F. Legoll, B. Stamm and S.Xiang, arxiv preprint 1807.05131], we have shown how to define threeapproximate homogenized diffusion coefficients on the basis of the embeddedcorrector problems. We have also proved that these approximations all convergeto the exact homogenized coefficients when the size of the bounded domainincreases.  We show here that, under the assumption that the diffusion matrix ispiecewise constant, the corrector problem to solve can be recast as an integralequation. In case of spherical inclusions with isotropic materials, we explainhow to efficiently discretize this integral equation using spherical harmonics,and how to use the fast multipole method (FMM) to compute the resultingmatrix-vector products at a cost which scales only linearly with respect to thenumber of inclusions. Numerical tests illustrate the performance of ourapproach in various settings.

An embedded corrector problem for homogenization. Part II: Algorithms and discretization