This paper uses the Modified Projection Method to examine the errors in solving the boundary integral equation from Laplace equation. The analysis uses weighted norms, and parallel algorithms help solve the independent linear systems. By applying the method developed by Kulkarni, the study shows how the approximate solution behaves in polygonal domains. It also explores computational techniques using the double layer potential kernel to solve Laplace equation in these domains. The iterated Galerkin method provides an approximation of order 2r+2 in smooth domains. However, the corners in polygonal domains cause singularities that reduce the accuracy. Adjusting the mesh near these corners can almost restore accuracy when the error is measured using the uniform norm. This paper builds on the work of Rude et al. By using modified operator suggested by Kulkarni, superconvergence in iterated solutions is observed. This leads to an asymptotic error expansion, with the leading term being $O(h^4)$ and the remaining error term $O(h^6)$, resulting in a method with similar accuracy.

翻译：本文采用修正投影法研究拉普拉斯方程边界积分方程求解中的误差问题。分析过程采用加权范数，并借助并行算法求解独立线性系统。通过应用Kulkarni提出的方法，本研究揭示了多边形区域中近似解的行为特性。同时探讨了利用双层势核在这些区域求解拉普拉斯方程的计算技术。迭代Galerkin方法在光滑区域可提供2r+2阶近似精度，但多边形区域的角点会产生奇异性，从而降低计算精度。当采用一致范数度量误差时，通过在角点附近调整网格划分，几乎可以完全恢复计算精度。本文基于Rude等人的研究工作，采用Kulkarni建议的修正算子，在迭代解中观察到超收敛现象。由此推导出渐近误差展开式，其主项为$O(h^4)$，剩余误差项为$O(h^6)$，最终获得具有同等精度的计算方法。