Error-Controlled Static Layered-Medium Green’s Function Computation via hp -Adaptive Spectral Differential Equation Approximation Method

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.