Regularization methods for ill-conditioned system of the integral equation of the first kind with the logarithmic kernel

Abstract

The occurring mechanism of the ill-conditioned system due to degenerate scale in the direct boundary element method (BEM) and the indirect BEM is analytically examined by using degenerate kernels. Five regularization techniques to ensure the unique solution, namely hypersingular formulation, method of adding a rigid body mode, rank promotion by adding the boundary flux equilibrium (direct BEM), CHEEF method and the Fichera's method (indirect BEM), are analytically studied and numerically implemented. In this paper, we examine the sufficient and necessary condition of boundary integral formulation for the uniqueness solution of 2D Laplace problem subject to the Dirichlet boundary condition. Both analytical study and BEM implementation are addressed. For the analytical study, we employ the degenerate kernel in the polar and elliptic coordinates to derive the unique solution by using five regularization techniques for any size of circle and ellipse, respectively. Full rank of the influence matrix in the BEM using Fichera's method for both ordinary scale and degenerate scale is also analytically proved. In numerical implementation, the BEM programme developed by NTOU/MSV group is employed to see the validity of the above formulation. Finally, the circular and elliptic cases are numerically demonstrated by using five regularization techniques. Besides, a general shape of a regular triangle is numerically implemented to check the uniqueness solution of BEM.