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The effective conductivity of random checkerboards. (English) Zbl 1211.78010
Summary: An algorithm is presented for the fast and accurate solution of the electrostatic equation on multi-component random checkerboards. It relies on a particular choice of integral equation, extended as to separate ill-conditioning due to singular fields in corners from ill-conditioning due to interaction of clusters of well-conducting squares at large distances. Two separate preconditioners take care of the two separate phenomena. In a series of numerical examples, effective conductivities are computed for random checkerboards containing up to $10^{4}$ squares with conductivity ratios of up to $10^{6}$. The achievable relative precision in these examples is on the order of $10^{ - 11}$.

78A30Electro- and magnetostatics
78M25Numerical methods in optics
45B05Fredholm integral equations
65F08Preconditioners for iterative methods
65F22Ill-posedness, regularization (numerical linear algebra)
Full Text: DOI
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