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 Pp+1−1−Pp−1 finite element pair for the flux variable and the scattered field with p≥0. We can get optimal order convergence for the flux variable in both Hdiv-like norm and L2 norm and the scattered field in L2 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.
Highlights
Is an open and bounded domain, and f ∈ L2(Ω) represents a harmonic source. e Robin boundary condition (2) is the lowest-order absorbing boundary condition [1]. e applications of the Helmholtz equation are extensive in many practical applications, such as geophysics and radar detecting, simulation of ground penetrating, biomedical imaging, acoustic noise control, and seismic wave propagation. e numerical solution of the Helmholtz equation is fundamental to the simulation of time harmonic wave phenomena in acoustics, electromagnetics, and elasticity
It is well known that the discontinuous Galerkin methods are flexible and highly parallelizable, and discontinuous Galerkin methods are widely used to solve the Helmholtz equation numerically, such as interior penalty discontinuous Galerkin method [15], hybridizable discontinuous Galerkin method [16, 17], local discontinuous Galerkin method [18], and the references therein
Two local discontinuous Galerkin methods are studied in [18], where the P−11 − P−11 finite element pair was used to approximate the flux variable and the scattered field. ey obtain the suboptimal convergence for the flux variable
Summary
We consider the following nonhomogeneous Helmholtz equation with the Robin boundary condition:. It is well known that the discontinuous Galerkin methods are flexible and highly parallelizable, and discontinuous Galerkin methods are widely used to solve the Helmholtz equation numerically, such as interior penalty discontinuous Galerkin method [15], hybridizable discontinuous Galerkin method [16, 17], local discontinuous Galerkin method [18], and the references therein. Two local discontinuous Galerkin methods are studied in [18], where the P−11 − P−11 finite element pair was used to approximate the flux variable and the scattered field. We set the penalty parameter β 0 in numerical fluxes and use the discontinuous P−p1+1 − P−p1(p ≥ 0) finite element pair to approximate the flux variable and the scattered field.
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