## Abstract

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.

Original language | English (US) |
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Article number | 110000 |

Journal | Journal of Computational Physics |

DOIs | |

State | Accepted/In press - 2020 |

### Bibliographical note

Funding Information:F. V.-C., N.- C. N and J. P. acknowledge support from the AFOSR Grant No. FA9550-19-1-0240 . S.-H.O. acknowledge support from the NSF Grant No. ECCS 1809240 and ECCS 1809723 . F. V.-C. acknowledges Vimworks for the design of Fig. 4 (a).

## Keywords

- Hybridizable discontinuous Galerkin method
- Hydrodynamic model for metals
- Maxwell's equations
- Nonlinear plasmonics
- Nonlocal electrodynamics
- Second-harmonic generation