Linkage between the unit logarithmic capacity in the theory of complex variables and the degenerate scale in the BEM/BIEMs

It is well known that BEM/BIEM results in degenerate scale for a two-dimensional Laplace interior problem subjected to the Dirichlet boundary condition. In such a case, there is nontrivial boundary normal flux even if the trivial boundary potential is specified. It is proved that the unit logarithmic capacity in the Riemann conformal mapping with respect to the unit circle results in a null field for the interior domain. The logarithmic capacity is defined as the leading coefficient of the linear term in the Riemann conformal mapping. First, the real-variable BIE is transformed to the complex variable BIE. By considering the analytical field and taking care of the path of the branch cut, we can prove that unit logarithmic capacity in the Riemann conformal mapping results in a degenerate scale. When the logarithmic capacity is equal to one, a trivial interior field can be obtained but an exterior field is derived to be nonzero using the logarithmic function. Two mapping functions, the Riemann conformal mapping for the geometry and the logarithmic function for the physical field, are both utilized. This matches well with the BEM result that an interior trivial field yields nonzero boundary flux in case of degenerate scale. Regarding the ordinary scale, BIE results in a null field in the exterior domain owing to the Green's third identity. It is interesting to find that ordinary and degenerate scales result in a null field in the exterior and interior domains, respectively. A parameter study for the scaling constant and the leading coefficient of the z term in the Riemann conformal mapping is also done. To demonstrate this finding, different shapes were demonstrated. Theoretical derivation using the Riemann conformal mapping with the unit logarithmic capacity and the degenerate scale in the BEM/BIEM both analytically and numerically indicate the null field in the interior domain.