Obtaining the upper bound of discretization error and critical boundary integrals of circular arc boundary element method

The boundary element method (BEM) is a popular method of solving linear partial differential equations, and it can be applied in many areas of engineering and science including fluid mechanics, acoustics, electromagnetics. The circular arc element (CAE) method is a scheme to discretize the boundary of problems arising in the BEM. In Dehghan and Hosseinzadeh (2011) [10] the order of the convergence of CAE discretization was obtained for the 2D Laplace equation. The current work extends the formulation developed in Dehghan and Hosseinzadeh (2011) [10] to convert CAE to a robust discretization method. In the present paper the upper bound of the CAE’s discretization error is determined theoretically for the 2D Laplace equation. Also we present a new method based on the complex space CC to obtain CAE’s boundary integrals without facing singularity and near singularity. Since there is no efficient approach to treat the near singular integrals of CAE in the BEM literature, the new scheme presented in this paper enhances the CAE discretization significantly. Several test problems are given and the numerical simulations are obtained which confirm the theoretical results.