Given James A, Hwang Chi-Ok
Naval Research Laboratory, MS5720, 4555 Overlook Avenue, SW Washington, DC 20375, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Oct;68(4 Pt 2):046128. doi: 10.1103/PhysRevE.68.046128. Epub 2003 Oct 27.
Elliptic boundary value problems are difficult to treat in the vicinity of singularities, i.e., edges and corners, of the boundary. The concentration of electrical charge on the edges and corners of a conductor is perhaps the simplest example of such problems. Here we provide a rapid method for accurate treatment of these problems. It utilizes a Green's-function-based implementation of last-passage Monte Carlo diffusion methods. This is combined with a diffusion algorithm for the scaling of solutions to the Laplace equation near a corner singularity to yield the solution of a benchmark problem: the charge distribution near the edge and corner of a conducting cube.
椭圆型边值问题在边界的奇点（即边和角）附近很难处理。导体边缘和角上电荷的集中可能是这类问题最简单的例子。在此，我们提供一种快速方法来精确处理这些问题。它利用基于格林函数的末次通过蒙特卡罗扩散方法的实现。这与一种扩散算法相结合，用于在角奇点附近对拉普拉斯方程的解进行缩放，以得到一个基准问题的解：导电立方体边缘和角附近的电荷分布。
