A numerically robust and computationally efficient approach for evaluating a planar layered substrate’s static Green’s function is developed based on the adaptive form of the spectral differential equation approximation method. The method uses a $p$ th-order finite element method (FEM) solution of the 1-D ordinary differential equation governing the spectrum of the layered-media Green’s function with spatial $h$ -adaptive meshing. The resulting pole-residue form of the Green’s function spectrum enables analytic evaluation of the pertinent Sommerfeld integrals providing $O(h^{p})$ error control of the spatial layered-medium Green’s function in near, intermediate, and far zones. The detailed error analysis is presented enabling automation of the 1-D FEM mesh refinement, which guarantees a prescribed accuracy of the solution depending on the distance between the source and observation locations. The method is well suited for computing Green’s function databases used by method of moments capacitance and inductance extractors.

中文翻译：

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基于hp自适应谱微分方程逼近法的误差控制静态分层介质格林函数计算

基于光谱微分方程近似方法的自适应形式，开发了一种用于评估平面分层基板的静态格林函数的数值稳健且计算效率高的方法。该方法使用一个 $p$ 控制分层介质格林函数谱的一维常微分方程的三阶有限元法 (FEM) 解 $h$ - 自适应网格。格林函数谱的所得极点残差形式可以对相关的索末菲积分进行分析评估，提供 $O(h^{p})$ 近、中、远区域空间分层中格林函数的误差控制。提供了详细的误差分析，实现了 1-D FEM 网格细化的自动化，这保证了解决方案的规定精度，具体取决于源和观察位置之间的距离。该方法非常适合计算由电容和电感提取器的矩方法使用的格林函数数据库。