The Green's functions for the Laplace equation satisfying the Dirichlet and Neumann boundary conditions on the upper side of the infinite plane with a circular hole are introduced and studied. These functions enable solutions of the boundary value problems in domains where the hole is closed by an arbitrary mesh (locally rough surfaces). The developed approach accounts for arbitrary positive and negative ground elevations inside the domain of interest, which is not possible to achieve using the regular method of images. Such problems appear in electrostatics, however, the methods developed apply to other domains where the Laplace or Poisson equations govern. Integral and series representations of the Green's functions are provided. Using these Green's functions, an efficient computational technique based on the boundary element method with fast multipole acceleration is developed. A numerical study of some benchmark problems is presented.

介绍并研究了满足狄利克雷和诺依曼边界条件的拉普拉斯方程在具有圆孔的无限平面上侧的格林函数。这些函数可以解决由任意网格（局部粗糙表面）封闭孔的域中的边界值问题。开发的方法考虑了感兴趣域内的任意正负地面高程，这是使用常规图像方法无法实现的。这些问题出现在静电学中，然而，所开发的方法适用于拉普拉斯或泊松方程支配的其他领域。提供了格林函数的积分和级数表示。使用这些格林函数，开发了一种基于边界元法的快速多极加速计算技术。提出了一些基准问题的数值研究。