A Mixed Discontinuous Galerkin Method for the Helmholtz Equation

In this paper, we introduce and analyze a mixed discontinuous Galerkin method for the Helmholtz equation. &e mixed discontinuous Galerkin method is designed by using a discontinuous P 1 p+1 − P 1 p 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 H(div)-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.


Introduction
We consider the following nonhomogeneous Helmholtz equation with the Robin boundary condition: where d � 2, 3, i � �� � − 1 √ denotes the imaginary unit, k ∈ R + is a given positive number and known as the wave number, Ω is an open and bounded domain, and f ∈ L 2 (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. However, it remains a challenge to design robust and efficient numerical algorithms for the Helmholtz equation, especially with the large wave number or highly oscillatory solution.
In recent years, many numerical methods have been developed to solve and analyze the Helmholtz equation, for instance, finite difference method [2][3][4][5], conforming finite element method [6,7], boundary element method [8], weak Galerkin finite element method [9][10][11][12], spectral method [13], and adaptive finite element method [14] . It is also well known that the discontinuous Galerkin methods are flexible and highly parallelizable, and hence 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. However, the local discontinuous Galerkin method [19][20][21] is known to be more physical and flexible on designing discontinuous Galerkin schemes. Two local discontinuous Galerkin methods are studied in [18], where the P − 1 1 − P − 1 1 finite element pair was used to approximate the flux variable and the scattered field. ey obtain the suboptimal convergence for the flux variable. In the numerical fluxes, the authors choose the penalty parameter β > 0 in theoretical analysis and numerical experiments.
In this paper, we set the penalty parameter β � 0 in numerical fluxes and use the discontinuous P − 1 p+1 − P − 1 p (p ≥ 0) finite element pair to approximate the flux variable and the scattered field. In addition, we obtain the optimal convergence for the scattered field and the flux variable in the L 2 norm and the flux variable in the H(div)-like norm numerically. e paper is organized as follows. In Section 2, we give the definition of the numerical fluxes and introduce the mixed discontinuous Galerkin method. A series of numerical experiments are presented to validate the effectiveness of the mixed discontinuous Galerkin method in Section 3. e paper concludes with conclusions in Section 4.

Meshes and Notations.
In order to establish the mixed discontinuous Galerkin method, we first introduce some notations. e standard space, norm, and inner product notations are adopted in this paper. In particular, for a bounded domain D, (·, ·) D and 〈·, ·〉 zD denote the L 2 -inner product on complex-valued spaces L 2 (D) and L 2 (zD), respectively.
Let T h be a shape regular triangulation of the domain Ω with mesh size h, h K be the diameter of any K ∈ T h and h � max K∈T h h K , E h be the set of all edges in T h , E I h be the collection of all interior edges, and E B h � E h /E I h be the set of boundary edges. For an interior edge e shared by two elements K 1 and K 2 , n 1 and n 2 are the unit outward normal vectors of them, respectively. For any scalar function v and vector-valued function τ, let v i � v| zK i and τ i � τ| zK i (i � 1, 2), and the average · { } and jump [·] are defined as follows: [v] � v 1 n 1 + v 2 n 2 , [τ] � τ 1 · n 1 + τ 2 · n 2 , e ∈ E I h . (3) where n is the outward unit normal vector on zΩ.

Mixed Discontinuous Galerkin Scheme.
In this section, we introduce the mixed discontinuous Galerkin method for the nonhomogeneous Helmholtz equations (1) and (2). Firstly, we introduce an auxiliary variable σ � − Δu and rewrite equation (1) into Next, we introduce the mixed discontinuous Galerkin method to find the solution pair (u, σ) of (5) and (6) numerically. Multiplying (5) and (6) by test functions τ and v, respectively, and integrating both equations over an element where n K denotes the unit outward normal vector to zK and the overbar denotes complex conjugation. e mixed discontinuous Galerkin spaces V h and Q h are defined as where P r (K)(r ≥ 1) stands for the set of all polynomials of degree less than or equal to r on K. Based on the weak formulations (7) and (8), the solution (u h , σ h ) of the mixed discontinuous Galerkin method can be defined by where u K and σ K are the numerical fluxes on the boundary zK. e numerical fluxes have to be suitably defined in order to ensure the stability and accuracy of the mixed discontinuous Galerkin method. Summing the above equations over all elements, K ∈ T h , and using the following integration by parts identity, we get Here, for any piecewise smooth scalar function v and vector-valued function τ, let div h τ| K � divτ| K on any element K ∈ T h . In this paper, we choose the numerical fluxes in (12) and (13) as follows: where η e is positive constant. Substituting the numerical fluxes (14) and (15) into (12) and (13), we get the mixed discontinuous Galerkin formulation of (1) and (2) as follows:

Numerical Experiments
In this section, we present some numerical examples of the mixed discontinuous Galerkin method. In the following experiments, we set penalty parameter η e � 10.

Example 1: Helmholtz Equation in Convex Domain.
We first consider a Helmholtz problem defined on the domain Ω � [− 0.5, 0.5] × [− 0.5, 0.5], and the exact solution is given by where r � ������ x 2 + y 2 and J n (z) are the Bessel functions of the first kind and order n. e function g is chosen by the exact solution. We test uniformly triangular meshes as shown in Figure 1(a). e numerical errors and convergence rate for k � 5 and k � 10 with the discontinuous Galerkin pair Tables 1-4. We can observe that the convergence rate is optimal for the flux variable in both H(div)-like norm and L 2 norm and the scattered field in L 2 norm.

Example 2: Helmholtz Equation in Nonconvex Domain.
In this numerical test, we solve the Helmholtz problem defined on a nonconvex domain as shown in Figure 1(b). We take f � 0 and choose the exact solution as follows: e boundary condition g is given by the exact solution u for ξ � 1, ξ � 3/2, and ξ � 2/3. We can check that u is smooth for ξ ∈ N, while its derivative has a singularity at (− 1, 0) for ξ ∉ N. For ξ � 1, 3/2, and 2/3 and wave number k � 4, Figures 2 and 3 show the errors of the mixed discontinuous Galerkin method with the finite element pair P − 1 1 − P − 1 0 and P − 1 2 − P − 1 1 , respectively. It shows that convergent rates are optimal in three errors for cases ξ � 1. We lose the optimal convergent in the flux variable ‖σ − σ h ‖ 0,Ω , while the scalar variable ‖u − u h ‖ 0,Ω and the flux variable ‖div(σ − σ h )‖ 0,Ω still converge at the optimal order, for the case ξ � 2/3 and ξ � 3/2.

Example 3: Helmholtz Equation with Large Wave Number.
is test is devoted to studying the numerical test for the same setting as example 1 with a large wave number. We solve the Helmholtz problem with different mesh sizes for four wave numbers k � 5, 10, 50, and 100. e errors in ‖u − u h ‖ 0,Ω , ‖σ − σ h ‖ 0,Ω , and ‖div(σ − σ h )‖ 0,Ω are shown in Figure 4 with p � 0 and Figure 5 with p � 1. It indicates that the mixed discontinuous Galerkin method is convergent for the cases k � 5 and k � 10, and the errors begin to reduce as h becomes to be quite small for large wave numbers k � 50 and 100. e surface plots of the mixed discontinuous Galerkin solution and the exact solution are shown in Figure 6. It shows that the mixed discontinuous Galerkin solution has the correct shape.

Example 4: Helmholtz Equation with Perturbation.
We consider the Helmholtz equation with perturbation [14] in the following form: is the incident field and the perturbation function q(x) has compact support in the domain Ω. Note that the solution is propagating for q(x) > − 1, while it is evanescent for q(x) < − 1. e numerical fluxes for the Helmholtz equation with perturbation are the same as (14) and (15). In this test, let Ω � [− 1.           to the scattering of the plane wave impinged on the negative perturbation, the wave gradually disappears near the origin.

Conclusion
e paper developed the mixed discontinuous Galerkin method for the Helmholtz equation. We define the numerical fluxes and introduce the mixed Galerkin method.
Compared with [18], we set penalty parameter β � 0 in numerical fluxes and use the discontinuous P − 1 P+1 − P − 1 p (p ≥ 0) finite element pair to approximate the flux variable and the scattered field. We present several numerical experiments to validate the effectiveness of the mixed discontinuous Galerkin method. e optimal convergence for the scattered field and the flux variable in the L 2 norm and the flux variable in the H(div)-like norm is obtained numerically. Furthermore, we test the Helmholtz equation with a large wave number and the Helmholtz equation with perturbation function and the rectangular waveguide, which indicate the well performance of the mixed discontinuous Galerkin method.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.