An Analysis of HDG Methods for the Helmholtz Equation

The finite element method has been widely 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 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 build-in stabilization mechanisms and a reduced global linear system. In this paper, we study the HDG methods for the Helmholtz equation with 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 L2 norm for piecewise polynomial spaces with arbitrary degree.