Degenerate scale for multiply connected Laplace problems

The degenerate scale in the boundary integral equation (BIE) or boundary element method (BEM) solution of multiply connected problem is studied in this paper. For the mathematical analysis, we use the null-field integral equation, degenerate kernels and Fourier series to examine the solvability of BIE for multiply connected problem in the discrete system. Two treatments, the method of adding a rigid body term and CHEEF concept (Combined Helmholtz Exterior integral Equation Formulation), are applied to remedy the non-unique solution due to the critical scale. The efficiency and accuracy of the two regularizations are also addressed. For simplicity without loss of generality, the eccentric case is considered to demonstrate the occurring mechanism of degenerate scale.