  In this paper, we consider numerical simulations of the nonlocal optical
response of metallic nanostructure arrays inside a dielectric host, which is of
particular interest to the nanoplasmonics community due to many unusual
properties and potential applications. Mathematically, it is described by
Maxwell's equations with discontinuous coefficients coupled with a set of
Helmholtz-type equations defined only on the domains of metallic
nanostructures. To solve this challenging problem, we develop an efficient
multiscale method consisting of three steps. First, we extend the system into
the domain occupied by the dielectric medium in a novel way and result in a
coupled system with rapidly oscillating coefficients. A rigorous analysis of
the error between the solutions of the original system and the extended system
is given. Second, we derive the homogenized system and define the multiscale
approximate solution for the extended system by using the multiscale asymptotic
method. Third, to fix the inaccuracy of the multiscale asymptotic method inside
the metallic nanostructures, we solve the original system in each metallic
nanostructure separately with boundary conditions given by the multiscale
approximate solution. A fast algorithm based on the $LU$ decomposition is
proposed for solving the resulting linear systems. By applying the multiscale
method, we obtain the results that are in good agreement with those obtained by
solving the original system directly at a much lower computational cost.
Numerical examples are provided to validate the efficiency and accuracy of the
proposed method.
