论文标题
快速计算分段恒定电导率的静电电势
Fast Computation of Electrostatic Potentials for Piecewise Constant Conductivities
论文作者
论文摘要
我们提出了一种新的数值方法，用于解决静电势势的椭圆形偏微分方程问题，并具有分段恒定电导率。我们采用一种积分方程方法，通过将解决方案表示为单层电位的总和来得出一个良好条件的积分方程系统。只要图层分离良好，所产生的积分运算符的内核是平滑的。快速多重方法用于加速积分方程的广义最小残留方法解。为了效率，我们基于层电荷密度的光谱分辨率来适应Nyström方法的网格。此外，我们提出了一种评估在整个域中高效且准确的解决方案的方法，从而避免了近距离评估问题。为了支持数值方法的设计选择，我们从其区域之间边界的电导率和几何形状方面明确地得出了规律性估计。所得的方法是快速准确的，以求解具有分段恒定电导率的介质中的静电电势。
We present a novel numerical method for solving the elliptic partial differential equation problem for the electrostatic potential with piecewise constant conductivity. We employ an integral equation approach for which we derive a system of well-conditioned integral equations by representing the solution as a sum of single layer potentials. The kernel of the resulting integral operator is smooth provided that the layers are well-separated. The fast multiple method is used to accelerate the generalized minimal residual method solution of the integral equations. For efficiency, we adapt the grid of the Nyström method based on the spectral resolution of the layer charge density. Additionally, we present a method for evaluating the solution that is efficient and accurate throughout the domain, circumventing the close-evaluation problem. To support the design choices of the numerical method, we derive regularity estimates with bounds explicitly in terms of the conductivities and the geometries of the boundaries between their regions. The resulting method is fast and accurate for solving for the electrostatic potential in media with piecewise constant conductivities.
