This paper concerns the development of hybridizable discontinuous Galerkin (HDG) methods for the following time-harmonic Maxwell system of equations with the impedance boundary condition
where is a polyhedral domain, denotes the imaginary unit while denotes the unit outward normal to , and is the tangential component of the electric field . The real number is called the wave number and is known as the impedance constant.
The Maxwell system (1.1) is the governing equations of a fixed-frequency electromagnetic wave propagation in homogeneous media. In many scientific and engineering applications high frequency (i.e., large wave number κ) electromagnetic waves must be considered. A large wave number κ makes problem (1.1)–(1.2) become strongly indefinite. This and the non-Hermitian feature of the problem are the main sources of the difficulties for computing high frequency waves. In order to resolve a very oscillatory high frequency wave, one must use sufficiently many grid points in a wave length. The rule of thumb is to have 6 to 12 points in a wave length. Since the wave length is inversely proportional to the wave number, this implies that a very small mesh size must be used in the case of high frequency waves. Consequently, one has to solve huge algebraic systems which are non-Hermitian and strongly indefinite, regardless which discretization methodology is used. In addition, comparing with the high frequency scalar waves, whose governing equation is the well-known Helmholtz equation, the Maxwell system (1.1) is more difficult to solve for two main reasons. First, this is because (1.1) is a system of three equations in a three-dimensional domain Ω. Second, the operator has a much larger non-trivial kernel (than that of the gradient operator ), which makes analysis of Maxwell system (1.1) and its numerical analysis harder than those for the Helmholtz equation.
In recent years, several types of numerical methods have been developed for problem (1.1)–(1.2), including Nédélec edge element (or conforming finite element) methods [21, 15] and discontinuous Galerkin methods [12, 13, 19, 3]. Lately, Hiptmair et al.  proposed a Trefftz DG method for the homogeneous time-harmonic Maxwell equations with the impedance boundary condition. Optimal rates of convergence were also derived. Trefftz DG methods are non-polynomial finite element methods which use special trial and test spaces consisting of the local solutions of the underlying PDE (1.1). Recently, Feng and Wu  proposed and analyzed an interior penalty DG (IPDG) method for problem (1.1)–(1.2), which is stable and uniquely solvable without any mesh constraint. The IPDG method exhibits several advantages over the standard finite element methods, such as flexibilities in constructing trial and test spaces, allowing the use of unstructured meshes, freedom of tuning the penalty parameters to reduce the pollution error. On the other hand, the dimension of the IPDG space is much larger than that of the corresponding finite element space, which adds computational cost for solving the resulting algebraic system.
To remedy the drawback of the IPDG method, we consider to adapt the hybridizable discontinuous Galerkin (HDG) methodology to problem (1.1)–(1.2). To a large extent, this paper is an extension of , where a similar HDG method was developed and analyzed for the time-harmonic acoustic wave propagation governed by the scalar Helmholtz equation. These works are motivated by the benefit of HDG methods in retaining the advantages of IPDG (and local DG) methods with a significantly reduced degree of freedom. That is achieved by introducing a new variable on the element edges such that the solution inside each element can be computed in terms of the new variable and in parallel. Therefore, the resulting algebraic system is only for the unknowns on the skeleton of the mesh. Two HDG methods were proposed in  for the time-harmonic Maxwell equations with the Dirichlet boundary condition. The first HDG method explicitly imposes a divergence-free constraint on the electric field . The divergence-free constraint then introduces a Lagrange multiplier to the system and subsequently results in a mixed curl-curl formulation for the problem whose primary unknowns are the electric field and the Lagrange multiplier. The second HDG method in  does not explicitly enforce the divergence-free constraint and thus results in a primal formulation for the problem which solves for the electric field only. Two-dimensional numerical results were presented in  to show the performance and accuracy of both methods, however, convergence analysis and rate of convergence were not addressed in the paper.
The primary goals of this paper are to adapt the second HDG method of  to problem (1.1)–(1.2) and to provide a complete stability and convergence analysis for the HDG method. The highlights of our main results include (i) to establish the well-posedness of the HDG method for any and ; (ii) to derive the following stability and error estimates for all and :
where denotes our HDG numerical solution, and
We remark that according to the “rule of thumb” the mesh size h should satisfy the constraint . However, estimate (1.3) indicates that the errors are not completely controlled by the product and they grow in κ while is fixed. Our numerical experiments also support this theoretical prediction. So our analysis provides a proof of the existence and a quantification of so-called “pollution effect”. Moreover, from above the estimates we can easily obtain the following improved stability and error estimates in the mesh regime :
We note that these error bounds decrease in κ under the new mesh constraint.
The remainder of this paper is organized as follows: In Section 2 we introduce the DG notation and describe the formulation of our HDG method for problem (1.1)–(1.2). Section 3 is devoted to the stability estimate for our HDG method, it is followed by a complete error analysis in Section 4. This is done by using a non-standard two-steps argument adapted from . Firstly, we derive error estimates for the HDG approximation of an auxiliary problem. Secondly, we derive the desired error estimates based on the stability estimate and the estimates for the auxiliary problem. Finally, we present two numerical experiments in Section 5 to validate our theoretical results and to gauge the performance of the proposed HDG method.
2 Formulation of the HDG Method
Throughout this paper we use the standard function and space notation (see Adams ). For any space V, let . We also use and for the inequalities and , respectively, where C is a positive number independent of the mesh size and wave number κ, but it may take different values in different occurrences. For ease of the presentation, we suppose and .
Let be a bounded polyhedral domain and denote a partition of Ω consisting of shape regular tetrahedra. Let be the diameter of T and . The set of all faces of is denoted by , the set of all interior faces by , and the set of all boundary faces by . We also define the jump of on an interior face as
We introduce the following DG spaces over the mesh :
where . We also define the following inner products on and , respectively:
are the standard notation.
Based on the above mixed formulation, performing local integration by parts and using the concepts of numerical fluxes and hybridization, our HDG method is defined as finding such that
Here the numerical flux is defined as
and the constant is the so-called local stabilization parameter which has an important effect on both the stability and the accuracy of the HDG method. In this paper, we choose .
As already alluded in Section 1, one advantage of the above HDG method is that both and can be eliminated from the scheme. It then leads to a system for only. We refer the reader to  for the detailed explanation.
We conclude this section by recalling the approximation properties of the projection operators
which are defined by
The above approximation properties of and will play an important role in our convergence analysis to be given in Section 4.
3 Stability Analysis
The goal of this section is to derive a stability estimate for the HDG scheme (2.2)–(2.6) for all . First, we quote some regularity results for the solution to problem (1.1)–(1.2), their proofs can be found in [10, 8].
Suppose that and
Then the solution is in for , and the estimate holds, where
where are the components of the vector function , it can be shown that . Furthermore, if , it can also be shown that (cf. [8, Remark 2.3])
Second, to derive our stability result, we need the following lemma.
Next, we quote a well-known approximation result of the edge finite element method from [12, Proposition 4.5], which relates the DG space with the Nédélec edge finite element space of the second type.
For any , there exist and such that
Let be defined by
We introduce the following Helmholtz decomposition for :
where and are defined by
It is clear that the above decomposition is orthogonal in the -inner product and . By [12, Lemma 4.4], we have
We are now ready to state our stability result.
The main idea of the proof is to use a duality argument and the PDE stability estimate. Since the proof is long, we divide it into three steps.
Step 1: It follows from Lemma 3.4 that there exists such that
The above admits the Helmholtz decompositions
Step 2: Consider the auxiliary problem
which has a unique solution satisfying the estimate
By (3.17), Green’s formula and the definition of the -projection, we obtain
and the fact that (2.4) implies
Next, we derive an upper bound for the term . Notice that
Since is continuous across the inner faces, the above equality becomes
Step 3: Setting in (2.3) yields . Using Young’s inequality again, we get
Choosing δ small enough and using (3.21), we obtain
4 Convergence Analysis
4.1 Error Estimates for an Auxiliary Problem
In this section, we consider an auxiliary problem whose HDG approximation is defined as follows: Find such that
for all and , where
It is easy to see that is an HDG elliptic projection of . Indeed we have the following result.
By the definition of , we have
Substituting the above expression into (4.10) yields
Taking the absolute value on both sides of (4.11) and using Young’s inequality, we obtain
Next we use the duality argument to estimate the -norm of .
It follows from  that there exists such that
We introduce the Helmholtz decompositions for :
Moreover, the decomposition is orthogonal in the -inner product and satisfies
Now we consider the following auxiliary problem:
Similar to the proof in , it can be shown that there exists a unique such that
By Green’s formula and (4.19) we get
Thus, taking in (4.2) and using Green’s formula yield
Then by (4.21) and the fact that is continuous across the inner faces, we get
Using the definition of , we can rewrite as
Since , we have
which completes the proof. ∎
Clearly, the first term on the right-hand side is one half order lower than being optimal for the linear element. In the following we derive an improved error estimate for with the help of a better estimation for .
where denotes the set of all quadratic polynomials on T. Obviously, is a subspace of . The next lemma establishes both the Helmholtz decomposition and the discrete Helmholtz decomposition as well as their relationship for each .
For each , there exist two decompositions for :
where , , and . Moreover, both decompositions are orthogonal in the -inner product, is divergence-free, and the following error estimate holds:
From [2, Theorem 3.12], there exists a unique such that
Since , there exists such that , hence and the orthogonality of and in the -inner product is obvious.
Next, we define such that
Let denote the interpolation operator to the second type Nédélec edge finite element space and be the interpolation operator to the first type edge finite element space. It is well known [15, 16] that these two operators satisfy the commuting diagram property, i.e.,
Since and is piecewise constant, we get
which implies that there exists such that . Hence using (4.29), we get
The proof is complete. ∎
We now derive an improved estimate for in the mesh regime .
Also, by Lemma 4.4, we have
Similar to the derivation of the estimate for in Lemma 4.2, it can be shown that
substituting the above equality into (4.34) yields
By Young’s inequality, we have
The proof is complete. ∎
Now we are ready to state our optimal error estimate for .
4.2 Error Estimates for the HDG Method
The goal of this section is to derive error estimates for our HDG method (2.2)–(2.5). This will be done by utilizing the stability estimate obtained in Theorem 3.5 and the error estimates of the elliptic projection established in the previous section.
Moreover, if , then
for all and . Moreover, the numerical flux satisfies
Applying the stability estimate in Theorem 3.5 to the above formulation yields
5 Numerical Experiments
In this section, we present two three-dimensional numerical experiments to validate our convergence theory and gauge the performance of the proposed HDG method. In both experiments, we use shape regular tetrahedral meshes and set the local stabilization parameter . The implementation is done using the MATLAB software package iFEM (cf. ).
For fixed wave number κ, we first compute the relative errors and , respectively. The left graph in Figure 1 displays the relative errors of the HDG solution for , while the right graph shows the same errors for . We observe that the relative errors of the HDG solution stay around 100% when the mesh size is not small enough for the wave number κ, which confirms the prediction of our error estimates for the HDG method. Moreover, we see that the pollution errors always appear on the coarse meshes, but they disappear on the fine meshes.
We present the relative errors and for the HDG approximation under different mesh conditions in Figure 2. The left graph shows the relationship between the relative errors and the wave number κ under the mesh condition . We observe that the relative errors cannot be controlled by and increase in κ, which indicates the existence of the pollution error. The right graph of Figure 2 displays the relative errors of the HDG solution under the mesh condition . It shows that under this mesh condition, the relative errors do not increase in κ.
To see other features of the HDG method, we consider the problem with wave number and show the phase error of the HDG solutions. We focus on the line segment and observe the traces of the real part of the first component of the HDG solutions with mesh sizes and , respectively. The traces of the real part of the first component of the exact solution are also plotted in blue in Figure 3. We see that the shapes of the HDG solutions match the traces of the exact one very well under the mesh condition .
in the polar coordinates, where and is the Bessel function of the first kind. Although we assume that the right-hand side of equation (1.1) is a solenoidal current density, we find that the numerical results are also valid for the case .
The left graph in Figure 4 displays the relative errors and when the mesh condition is enforced. It shows that the relative errors can not be controlled by and increase in κ, an indication of the existence of the pollution error. The right graph in Figure 4 shows the same errors with the slightly stronger mesh condition . It clearly indicates that the relative errors can be controlled under this mesh condition.
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About the article
Published Online: 2016-06-01
Published in Print: 2016-07-01
Funding Source: National Natural Science Foundation of China
Award identifier / Grant number: 11401417
Award identifier / Grant number: 11225107
This work was supported by the Sino-German Science Center (grant id 1228) on the occasion of the Chinese-German Workshop on Computational and Applied Mathematics in Augsburg 2015. The work of the second author was supported by the National Natural Science Foundation of China (grant no. 11401417), the Program of Natural Science Research of Jiangsu Higher Education Institutions of China (grant no. 14KJB110021), and the Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems (grant no. 201404). The work of the third author was supported by the National Natural Science Foundation of China (grant no. 11225107).