We develop a nested hybridizable discontinuous Galerkin (HDG) method to numerically solve the Maxwell's equations coupled with a hydrodynamic model for the conduction-band electrons in metals. The HDG method leverages static condensation to eliminate the degrees of freedom of the approximate solution defined in the elements, yielding a linear system in terms of the degrees of freedom of the approximate trace defined on the element boundaries. This article presents a computational method that relies on a degree-of-freedom reordering such that the HDG linear system accommodates an additional static condensation step to eliminate a large portion of the degrees of freedom of the approximate trace, thereby yielding a much smaller linear system. For the particular metallic structures considered in this article, the resulting linear system obtained by means of nested static condensations is a block tridiagonal system, which can be solved efficiently. We apply the nested HDG method to compute second harmonic generation on a triangular coaxial periodic nanogap structure. This nonlinear optics phenomenon features rapid field variations and extreme boundary-layer structures that span a wide range of length scales. Numerical results show that the ability to identify structures which exhibit resonances at ω and 2ω is essential to excite the second harmonic response.

我们开发了一种嵌套的可杂交不连续伽勒金（HDG）方法，以数值求解麦克斯韦方程组以及金属中导带电子的流体动力学模型。HDG方法利用静态冷凝来消除元素中定义的近似解的自由度，从而根据元素边界上定义的近似迹线的自由度产生线性系统。本文提出了一种计算方法，该方法依赖于自由度重排序，这样HDG线性系统可容纳一个额外的静态冷凝步骤，以消除近似迹线的大部分自由度，从而产生一个更小的线性系统。对于本文考虑的特定金属结构，通过嵌套的静态缩合获得的线性系统是块三对角线系统，可以有效地对其进行求解。我们应用嵌套的HDG方法来计算三角形同轴周期纳米间隙结构上的二次谐波产生。这种非线性光学现象具有快速的场变化和跨很大长度范围的极端边界层结构的特征。数值结果表明，能够识别在以下位置出现共振的结构 这种非线性光学现象具有快速的场变化和跨很大长度范围的极端边界层结构的特征。数值结果表明，能够识别在以下位置出现共振的结构 这种非线性光学现象具有快速的场变化和跨很大长度范围的极端边界层结构的特征。数值结果表明，能够识别在以下位置出现共振的结构ω和2ω对于激发二次谐波响应至关重要。