Analysis of a hybridizable discontinuous Galerkin method for the Maxwell operator
This work provides a theoretical foundation for an HDG method for Maxwell's equations, which is important for computational electromagnetics, but the results are incremental as they extend existing HDG analysis to a specific operator.
The paper analyzes a hybridizable discontinuous Galerkin method for the Maxwell operator, proving stability and optimal-order convergence for both high and low regularity cases, with numerical experiments confirming the theory.
In this paper, we study a hybridizable discontinuous Galerkin (HDG) method for the Maxwell operator. The only global unknowns are defined on the inter-element boundaries, and the numerical solutions are obtained by using discontinuous polynomial approximations. The error analysis is based on a mixed curl-curl formulation for the Maxwell equations. Theoretical results are obtained under a more general regularity requirement. In particular for the low regularity case, special treatment is applied to approximate data on the boundary. The HDG method is shown to be stable and convergence in an optimal order for both high and low regularity cases. Numerical experiments with both smooth and singular analytical solutions are performed to verify the theoretical results.