We present a numerical algorithm for an accurate and efficient computation of the convolution of the frequency domain layered media Green's function with a given density function. Instead of compressing the convolution matrix directly as in the classical fast multipole, fast direct solver, and $\mathcal{H}$-matrix algorithms, this new algorithm considers a translated form of the original matrix so existing blocks from the highly optimized free-space fast multipole method can be easily adapted to the layered media Green's function. An asymptotic analysis is performed on the Sommerfeld integrals to provide an estimate of the decay rate in the new “multipole” and “local” expansions. To avoid the highly oscillatory integrand in the original integral representations when the source and target are close to each other, mathematically equivalent alternative direction integral representations are introduced. The convergence of the new expansions and quadrature rules for the original and alternative direction representations are numerically validated.

我们提出了一种数值算法，用于精确有效地计算具有给定密度函数的频域分层媒体格林函数的卷积。而不是像传统的快速多极，快速直接求解器那样直接压缩卷积矩阵，$\mathcal{H}$-矩阵算法，此新算法考虑了原始矩阵的转换形式，因此高度优化的自由空间快速多极子方法中的现有块可以轻松地适应分层媒体Green的功能。对Sommerfeld积分进行渐近分析，以估计新的“多极”和“局部”展开中的衰减率。为了避免当源和目标彼此靠近时原始积分表示中的振荡整数，引入了数学上等效的替代方向积分表示。数值验证了原始方向和替代方向表示的新扩展和正交规则的收敛性。