An absolutely stable $hp$-HDG method for the time-harmonic Maxwell equations with high wave number

TL;DR

This paper introduces an absolutely stable hybridizable discontinuous Galerkin (HDG) method for solving time-harmonic Maxwell equations with high wave numbers, ensuring stability and convergence through duality analysis and verified by numerical experiments.

Contribution

The paper develops a new HDG method that guarantees absolute stability for high wave number Maxwell equations, with a detailed convergence analysis.

Findings

The HDG method is stable for high wave numbers.

Convergence depends on wave number, mesh size, and polynomial order.

Numerical results confirm theoretical predictions.

Abstract

We present and analyze a hybridizable discontinuous Galerkin (HDG) method for the time-harmonic Maxwell equations. The divergence-free condition is enforced on the electric field, then a Lagrange multiplier is introduced, and the problem becomes the solution of a mixed curl-curl formulation of the Maxwell's problem. The method is shown to be an absolutely stable HDG method for the indefinite time-harmonic Maxwell equations with high wave number. By exploiting the duality argument, the dependence of convergence of the HDG method on the wave number k, the mesh size h and the polynomial order p is obtained. Numerical results are given to verify the theoretical analysis.