Consider the Dirichlet problem for Laplace’s/Poisson’s equation in a bounded simply-connected domain $S$. The source function $q\ln |\overline{P{Q}^{*}}|$ is a fundamental solution (FS), and it can be found in many physical problems. The singularity occurs when the boundary value data affected by $q\ln |\overline{P{Q}^{*}}|$ as the source node $Q^{*}$ is located near the boundary $\Gamma (=\partial S)$. So far, there is no comprehensive study on this kind of singularity. In this paper, the solution singularity is explored and the reduced convergence rates are derived for the method of particular solutions (MPS) and the method of fundamental solutions (MFS). Classic domains, such as disks, ellipses and polygons, are discussed for analysis and computation. For this new kind of solution singularity, the convergence rates of the MFS and the MPS are very low. The errors caused by numerical integration are critical to the solution accuracy. A new analytic framework for the collocation Trefftz method (CTM) involving numerical integration is established in this paper; this is an advanced development of our previous study [19]. Since the numerical solutions are poor in accuracy, removal techniques are essential in applications. New removal techniques are proposed for a node $Q^{*}$ located near $\Gamma$. In this paper, an additional FS as, $d_0\ln \left|\overline{P{Q}_0}\right|$, is added to the original source nodes in the traditional MFS, and the point charge $d_0(=q)$ and the source node $Q_0$ are unknowns to be sought by nonlinear solvers (such as the secant method). When the source node $Q^{*}$ is located inside $S$ but near $\Gamma$, both simple domains (such as disks, ellipses and squares) and complicated domains (such as amoeba-like domains) are studied. The validity of the new removal techniques is supported by numerical experiments. The removal techniques in this paper may also be applied to solve source identification problems. A comprehensive study has been completed in this paper for the solitary source function $q\ln |\overline{P{Q}^{*}}|$ as the source node $Q^{*}$ is located near $\Gamma$.

考虑有界单连通域中拉普拉斯/泊松方程的狄利克雷问题 $秒$. 源函数$q\mathrm{输入}|\overline{磷{问}^{*}}|$是一个基本解（FS），它可以在许多物理问题中找到。当边界值数据受$q\mathrm{输入}|\overline{磷{问}^{*}}|$ 作为源节点 ${问}^{*}$ 位于边界附近 $\Gamma (=\partial 秒)$. 迄今为止，还没有对这种奇点进行全面的研究。在本文中，探讨了解的奇异性，并推导出了特解法（MPS）和基本解法（MFS）的收敛速度降低。讨论了经典域，例如圆盘、椭圆和多边形，以进行分析和计算。对于这种新的解奇点，MFS 和 MPS 的收敛速度非常低。数值积分引起的误差对求解精度至关重要。本文建立了一个新的包含数值积分的搭配Trefftz方法(CTM)的分析框架；这是我们之前研究的进一步发展 [19]。由于数值解的精度较差，去除技术在应用中是必不可少的。${问}^{*}$ 位于附近 $\Gamma$. 在本文中，一个额外的 FS 作为，$d_0\mathrm{输入}\left|\overline{磷{问}_0}\right|$, 在传统 MFS 中添加到原始源节点，点电荷 $d_0(=q)$ 和源节点 ${问}_0$是非线性求解器（例如割线法）要寻找的未知数。当源节点${问}^{*}$ 位于里面 $秒$ 但近 $\Gamma$，研究了简单域（如圆盘、椭圆和正方形）和复杂域（如变形虫状域）。数值实验支持新去除技术的有效性。本文中的去除技术也可用于解决源识别问题。本文完成了对孤源函数的综合研究$q\mathrm{输入}|\overline{磷{问}^{*}}|$ 作为源节点 ${问}^{*}$ 位于附近 $\Gamma$.