A regularization of the parametric integral equation system applied to 2D boundary problems for Laplace?s equation with stability evaluation

摘要

The paper proposes the regularization of the parametric integral equation system (PIES) for 2D potential problems governed by Laplace's equation. The regularization eliminates all singularities (weak and strong) in PIES associated with its kernels as a result of their modification and is based on introducing a regularizing function with unknown regularization coefficients. Hence, all integrals appearing in the regularized PIES formulation are non-singular and can be evaluated numerically with a standard low order Gauss-Legendre quadrature rule. Using this formulation, the stability of solutions for different distances between collocation points and quadrature nodes is investigated. The influence of the number of integration and collocation points on the accuracy and stability of solutions for the proposed regularization in comparison with dedicated procedures for evaluation of strongly and weakly singular integrals is also studied. Furthermore, we analyze the impact of the regularization in PIES on the results compared with other methods for evaluating singular integrals in nonregularized PIES.