Revisit of logarithmic capacity of line segments and double-degeneracy of BEM/BIEM

Abstract

Degenerate scale of line segments is derived in this paper. It is found that two formulas in the Rumely's book are inconsistent after comparing with our results. We confirm our formulas not only by testing the value of degenerate scale in the BEM/BIEM but also by finding the unit logarithmic capacity. Double degeneracy due to the degenerate boundary (line segment) and degenerate scale in the BEM/BIEM is also examined. By increasing the number of boundary elements, the value of degenerate scale converges to the corresponding one derived by using the complex variables instead of Rumely's formulas. Three cases in the Rumely's book, include one segment, two equal segments and three segments (L, 2 L, L) with the spacing 3 L, are re-confirmed. However, we found that there are two inconsistent cases, two unequal segments (L, 5 L) with the spacing 3 L and three segments (L, 2 L, L) with the spacing L, respectively, after analytically deriving by using the complex variables and numerically examining by detecting the degenerate scale in the BEM/BIEM.