High-order multiscale hybridizable discontinuous Galerkin method for a class of one-dimensional oscillatory second-order equations

In this paper, we propose and analyze multiscale hybridizable discontinuous Galerkin (HDG) methods for solving a class of second-order equations with oscillatory solutions in one dimension. The high-order multiscale finite element spaces $E^p$ and $T^{2p+1}$ contain non-polynomial basis functions that incorporate fine-scale features, as developed in our previous work (Dong and Wang, 2020). We prove that the resulting multiscale HDG method with the $E^p$ space achieves optimal convergence in both the primary variable and its derivative with respect to the mesh size $h$, provided $h$ is sufficiently small. Numerical experiments demonstrate that for both $E^p$ and $T^{2p+1}$ spaces, the multiscale HDG methods exhibit second-order convergence without any resonance errors even when $h$ is comparable to or larger than the wavelength scale, whereas the traditional HDG method with polynomial basis functions fails to converge in this regime. When $h$ is smaller than the scale of the wavelength, numerical results confirm the optimal high-order convergence predicted by the error analysis. Our numerical results also demonstrate that the proposed methods are capable of capturing highly oscillating solutions of the Schrödinger equation in the application of the resonant tunneling diode (RTD) model.