Near corner boundary regularization of the parametric integral equation system (PIES)

Abstract

This paper presents the regularization of the parametric integral equation system (PIES) and the integral identity for 2D problems governed by the Laplace equation in polygonal and smooth domains. Due to the singularity of PIES and the existence of near-singularities in the integral identity, significant errors occur in the results near the boundary, even in the case of smooth boundaries. Additionally, polygonal domains have corner points where the boundary is not smooth, resulting in even greater errors when finding solutions within the domain near these points. These errors increase significantly as the distance to the corner points decreases. The purpose of the presented regularization is to eliminate singularities in PIES and nearly singularities in the integral identity that occur near the boundary, thus improving the accuracy of solutions. The main idea of the regularization is to introduce a regularizing function with two unknown regularization coefficients that need to be determined. As a result, the modified PIES and the integral identity become nonsingular. The paper includes several numerical examples with boundaries described by Bézier and NURBS curves that demonstrate the reliability of the regularization approach and highlight significant improvements in accuracy near the boundary and corner points.