A New Method for Numerical Solution of Checkerboard Fields

for a standard checkerboard of conductivity λg and λw. The explicit solution of the corresponding temperature-field was later found by Berdichevskii [1]. In particular he found that the heat-flux is infinitely high in the corners of the squares. Subsequently, explicit solutions for rectangular and triangular checkerboards were found in [17, 19, 20]. Mortola and Steffé [18] presented in 1985 an interesting conjecture concerning the effective conductivity of four phase checkerboards. Many attempts were made to prove/disprove the conjecture, even by specialists in homogenization theory (see [22]), but the problem remained unsolved for the rest of the century. Very recently the conjecture has been proved by Craster and Obnosov [5] and independently by Milton [16] (using a completely different proof). There are many interesting works on other types of checkerboards. Concerning three-dimensional checkerboards, we refer to [12, 15] (see also [14]).