标题： 边界积分方程近似解的渐近展开 显示英文标题

标题： Asymptotic expansions for approximate solutions of boundary integral equations

摘要： 本文使用改进的投影方法来检查从拉普拉斯方程求解边界积分方程中的误差。 分析使用加权范数，并行算法有助于求解独立的线性系统。 通过应用Kulkarni开发的方法，研究展示了近似解在多边形区域中的行为。 它还探讨了使用双层势核求解这些区域中拉普拉斯方程的计算技术。 迭代伽辽金方法在光滑区域中提供了2r+2阶的近似。 然而，多边形区域中的角点会引起奇异性，从而降低精度。 当使用均匀范数测量误差时，在这些角点附近调整网格可以几乎恢复精度。 本文建立在Rude等人工作的基础上。 通过使用Kulkarni提出的修正算子，观察到迭代解中的超收敛现象。 这导致了一个渐近误差展开式，主要项为$O(h^4)$，其余误差项为$O(h^6)$，从而得到一种具有相似精度的方法。 显示英文摘要

摘要： 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.