Artificial Boundary Conditions for Random Elliptic Systems with Correlated Coefficient Field

. We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale \ (\) in an infinite heterogeneous correlated random medium, in a situation where the medium is only known in a box of diameter \ (L \) around the support of the charge. We show that the algorithm in J. Lu, F. Otto, and L. Wang, Optimal Artificial Boundary Conditions Based on Second-Order Correctors for Three Dimensional Random Ellilptic Media, preprint, arXiv: 2109. 01616, 2021, suggesting optimal Dirichlet boundary conditions motivated by the multipole expansion P. Bella, A. Giunti, and F. Otto, Comm. Partial Differential Equations, 45 (2020), pp. 561–640, still performs well in correlated media. With overwhelming probability, we obtain a convergence rate in terms of \ (\), \ (L\), and the size of the correlations for which optimality is supported with numerical simulations. These estimates are provided for ensembles which satisfy a multiscale logarithmic Sobolev inequality, where our main tool is an extension of the semigroup estimates in N. Clozeau, Stoch. Partial Differ. Equ. Anal. Comput. , 11 (2023), pp. 1254–1378. As part of our strategy, we construct sublinear second-order correctors in this correlated setting, which is of independent interest. Keywordsartificial boundary conditioncorrelated random mediastochastic homogenizationmultipole expansionMSC codes35B2765N99