This paper proposes the Boundary Knot Method with ghost points (BKM-G), which enhances the performance of the BKM for solving 2D (3D) high-order Helmholtz-type partial differential equations in domains with multiple cavities. The BKM-G differs from the conventional BKM by relocating the source points from the boundary collocation nodes to a random region, such as a circle (sphere) encompassing the original domain in 2D (3D). Compared with classical BKM, this modification improves accuracy without sacrificing simplicity and efficiency. Moreover, this paper investigates and analyzes the effect of the ghost circle/sphere’s radius R in BKM-G for solving various high-order Helmholtz-type PDEs. Numerous 2D and 3D numerical examples illustrate that the BKM-G outperforms the BKM for a wide range of R. The effectiveness of the proposed effective condition number (ECN) approach in finding the optimal R has also been demonstrated. Furthermore, the economic ECN (EECN) is studied to significantly improve the efficiency of ECN.

中文翻译：

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在多连通域中使用高阶亥姆霍兹型 PDE 的重影点的边界结法


本文提出了带鬼点的边界结法 （BKM-G），该方法增强了 BKM 在多腔域中求解 2D （3D） 高阶亥姆霍兹型偏微分方程的性能。BKM-G 与传统 BKM 的不同之处在于，它将源点从边界配置节点重新定位到随机区域，例如在 2D （3D） 中包围原始域的圆（球体）。与传统 BKM 相比，这种修改在不牺牲简单性和效率的情况下提高了准确性。此外，本文调查和分析了 BKM-G 中鬼圆/球体半径 R 对求解各种高阶亥姆霍兹型偏微分方程的影响。大量 2D 和 3D 数值示例表明，BKM-G 在较宽的 R 范围内优于 BKM。所提出的有效条件数 （ECN） 方法在寻找最佳 R 方面的有效性也得到了证明。此外，经济 ECN （EECN） 被研究为显着提高 ECN 的效率。