Optimal Artificial Boundary Condition for Random Elliptic Media

We are given a uniformly elliptic coefficient field that we regard as realization of stationary and finite-range ensemble fields. Given right-hand side supported in ball size $$\ell \gg 1$$ vanishing average, interested an algorithm to compute the solution near origin, just using knowledge some large box $$L\gg \ell $$ . More precisely, most seamless artificial boundary condition on computational domain L. Motivated by recently introduced multipole expansion random media, propose algorithm. rigorously establish error estimate level gradient terms , recent results quantitative stochastic homogenization. our has priori posteriori aspect: with overwhelming probability, prefactor can be bounded constant is computable without much further effort, basis also order both L optimal, where this paper focus case $$d=2$$ This amounts lower bound variance quantity interest when conditioned coefficients inside domain, relies deterministic insight sensitivity analysis respect defect commutes Finally, carry out numerical experiments show optimal convergence rate already sets at only moderately L, more naive conditions perform worse prefactor.