The finite element method has widely been used to discretize the Helmholtz equation with various types of boundary conditions. The strong indefiniteness of the Helmholtz equation makes it difficult to establish stability estimates for the numerical solution. In particular, discontinuous Galerkin methods for the Helmholtz equation with a high wave number result in very large matrices since they typically have more degrees of freedom than conforming methods. However, hybridizable discontinuous Galerkin (HDG) methods offer an attractive alternative because they have built-in stabilization mechanisms and a reduced global linear system. In this paper, we study HDG methods for the Helmholtz equation with a first-order absorbing boundary condition in two and three dimensions. We prove that the proposed HDG methods are stable (hence well posed) without any mesh constraint. The stability constant is independent of the polynomial degree. By using a projection-based error analysis, we also derive the error estimates in the L₂ norm for piecewise polynomial spaces with arbitrary degree.

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.