Variable Transformations for Nearly Singular Integrals in the Boundary Element Method

The Boundary Element Method (BEM) or the Boundary Integral Equation (BIE) method is a convenient method for solving partial differential equations, in that it requires discretization only on the boundary of the domain [2]. In the method, the accurate and efficient computation of boundary integrals is important. In particular, the evaluation of nearly singular integrals, which occur when computing field values near the boundary or treating thin structures, is not an obvious task. For this purpose, Lachat and Watson [25] proposed an adaptive element subdivision method using an error estimator for the numerical integration. Later, a more sophisticated variable order composite quadrature with exponential convergence was proposed by Schwab [27]. A different approach using quadratic and cubic variable transformations in order to weaken the near singularity before applying Gauss quadrature was introduced by Telles [29]. Koizumi and Utamura [20, 21] used polar coordinates with corrections. Hackbusch and Sauter [7] also used local polar coordinates, performing the inner integrals analytically and the outer integral by Gauss quadrature. Another approach is to subtract out the near singularity using analytical integration formulas for constant planar elements, and then evaluating the