A Mixed Discontinuous Galerkin Method for the Helmholtz Equation

Abstract

In this paper, we introduce and analyze a mixed discontinuous Galerkin method for the Helmholtz equation. The mixed discontinuous Galerkin method is designed by using a discontinuous ${\mathbf{P}}_{p+1}^{-1}-P_p^{-1}$ finite element pair for the flux variable and the scattered field with $p\ge 0$. We can get optimal order convergence for the flux variable in both $H\left(\text{div}\right)$-like norm and $L^2$ norm and the scattered field in $L^2$ norm numerically. Moreover, we conduct the numerical experiments on the Helmholtz equation with perturbation and the rectangular waveguide, which also demonstrate the good performance of the mixed discontinuous Galerkin method.