A nested hybridizable discontinuous Galerkin method for computing second-harmonic generation in three-dimensional metallic nanostructures

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. •A nested HDG method to economize the solution of traditional HDG systems.•A computational framework combining nested HDG with the structure of the problem for additional computational savings.•Model order reduction strategy to identify geometries that lead to resonances at ω, 2ω.•Ability to simulate nonlocal second harmonic generation plasmonic phenomena on realistic 3-D nanostructures.