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An integral equation based scheme is presented for the fast and accuratecomputation of effective conductivities of two-component checkerboard-likecomposites with complicated unit cells at very high contrast ratios. The schemeextends recent work on multi-component checkerboards at medium contrast ratios.General improvement include the simplification of a long-range preconditioner,the use of a banded solver, and a more efficient placement of quadraturepoints. This, together with a reduction in the number of unknowns, allows for asubstantial increase in achievable accuracy as well as in tractable systemsize. Results, accurate to at least nine digits, are obtained for randomcheckerboards with over a million squares in the unit cell at contrast ratio10^6. Furthermore, the scheme is flexible enough to handle complex valuedconductivities and, using a homotopy method, purely negative contrast ratios.Examples of the accurate computation of resonant spectra are given.

The effective conductivity of arrays of squares: large random unit cells and extreme contrast ratios