A Hybridizable Discontinuous Galerkin Method for the Time-Harmonic Maxwell Equations with High Wave Number

Abstract This paper proposes and analyzes a hybridizable discontinuous Galerkin (HDG) method for the three-dimensional time-harmonic Maxwell equations coupled with the impedance boundary condition in the case of high wave number. It is proved that the HDG method is absolutely stable for all wave numbers κ > 0 ${\kappa>0}$ in the sense that no mesh constraint is required for the stability. A wave-number-explicit stability constant is also obtained. This is done by choosing a specific penalty parameter and using a PDE duality argument. Utilizing the stability estimate and a non-standard technique, the error estimates in both the energy-norm and the 𝐋 2 ${\mathbf{L}^{2}}$ -norm are obtained for the HDG method. Numerical experiments are provided to validate the theoretical results and to gauge the performance of the proposed HDG method.

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