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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

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Volume 16, Issue 3

Issues

A Hybridizable Discontinuous Galerkin Method for the Time-Harmonic Maxwell Equations with High Wave Number

Xiaobing Feng / Peipei Lu / Xuejun Xu
  • Corresponding author
  • LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing, 100190; and Department of Mathematics, Tongji University, Shanghai, P. R. China
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Published Online: 2016-06-01 | DOI: https://doi.org/10.1515/cmam-2016-0021

Abstract

This paper proposes and analyzes a hybridizable discontinuous Galerkin (HDG) method for the three-dimensional time-harmonic Maxwell equations coupled with the impedance boundary condition in the case of high wave number. It is proved that the HDG method is absolutely stable for all wave numbers κ>0 in the sense that no mesh constraint is required for the stability. A wave-number-explicit stability constant is also obtained. This is done by choosing a specific penalty parameter and using a PDE duality argument. Utilizing the stability estimate and a non-standard technique, the error estimates in both the energy-norm and the 𝐋2-norm are obtained for the HDG method. Numerical experiments are provided to validate the theoretical results and to gauge the performance of the proposed HDG method.

Keywords: Hybridizable Discontinuous Galerkin Method; Time-Harmonic Maxwell Equations; High Wave Number; Error Estimates

MSC 2010: 65N12; 65N15; 65N30; 78A40

1 Introduction

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

curlcurl𝐄-κ2𝐄=𝐟~in Ω,(1.1)curl𝐄×𝐧-𝐢κλ𝐄T=𝐠~on Ω,(1.2)

where Ω3 is a polyhedral domain, 𝐢=-1 denotes the imaginary unit while 𝐧 denotes the unit outward normal to Ω, and 𝐄T:=(𝐧×𝐄)×𝐧 is the tangential component of the electric field 𝐄. The real number κ>0 is called the wave number and λ>0 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 curl 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. [11] 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 [8] 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 [6], 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 [18] 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 𝐄. 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 𝐄 and the Lagrange multiplier. The second HDG method in [18] does not explicitly enforce the divergence-free constraint and thus results in a primal formulation for the problem which solves for the electric field 𝐄 only. Two-dimensional numerical results were presented in [18] 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 [18] 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 κ>0 and h>0; (ii) to derive the following stability and error estimates for all κ>0 and h>0:

𝐄h0,Ω+𝐐h0,ΩCstab𝐟0,Ω+Cstab𝐠0,Ω,𝐄-𝐄h0,Ω(κh2+Cstabκ2h2)M(𝐟~,𝐠~),κ𝐐-𝐐h0,Ω(κh+Cstabκ3h2)M(𝐟~,𝐠~),(1.3)

where (𝐄h,𝐐h) denotes our HDG numerical solution, and

𝐐:=𝐢κcurl𝐄,𝐟:=𝐢κ𝐟~,𝐠:=-𝐢κ𝐠~,Cstab:=1+κ3h2+κ-32h-12,M(𝐟~,𝐠~):=𝐟~0,Ω+𝐠~1/2,Ω.

We remark that according to the “rule of thumb” the mesh size h should satisfy the constraint κh1. However, estimate (1.3) indicates that the errors are not completely controlled by the product κh and they grow in κ while κh 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 κ3h21:

𝐄h0,Ω+𝐐h0,Ω𝐟0,Ω+𝐠0,Ω,𝐄-𝐄h0,Ω(κh2+κ2h2)M(𝐟~,𝐠~),κ𝐐-𝐐h0,Ω(κh+κ3h2)M(𝐟~,𝐠~).

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 [8]. 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 [1]). For any space V, let 𝐕:=[V]3. We also use AB and AB for the inequalities ACB and ACB, 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 λ1 and κ1.

Let Ω3 be a bounded polyhedral domain and 𝒯h denote a partition of Ω consisting of shape regular tetrahedra. Let hT be the diameter of T and h:=maxT𝒯hhT. The set of all faces of 𝒯h is denoted by h, the set of all interior faces by hI, and the set of all boundary faces by hB. We also define the jump 𝐯 of 𝐯 on an interior face =T+T- as

𝐯|:={𝐯|T+-𝐯|T-if the global label of T+ is bigger,𝐯|T--𝐯|T+if the global label of T- is bigger.

We introduce the following DG spaces over the mesh 𝒯h:

𝐕h:={𝐯𝐋2(Ω):𝐯|T(P1(T))3 for all T𝒯h},𝐌h:={𝝁𝐋2(h):𝝁|F(P1(F))3 for all Fh},

where 𝐋2(h):=Fh(L2(F))3. We also define the following inner products on 𝐋2(h) and 𝐋2(Ω), respectively:

(𝐯,𝐰)𝒯h:=T𝒯h(𝐯,𝐰)Tfor all 𝐯,𝐰𝐋2(Ω),𝐯,𝐰𝒯h:=T𝒯h𝐯,𝐰Tfor all 𝐯,𝐰𝐋2(h).

Here

(𝐯,𝐰)T:=T𝐯𝐰¯𝑑xand𝐯,𝐰T:=T𝐯𝐰¯𝑑s

are the standard notation.

To define our HDG method, we rewrite (1.1)–(1.2) as the following mixed-form first-order system which seeks (𝐐,𝐄) such that

𝐢κ𝐐+curl𝐄=0in Ω,(2.1)curl𝐐-𝐢κ𝐄=𝐟in Ω,𝐧×𝐐-λ𝐄T=𝐠on Ω.

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 (𝐐h,𝐄h,𝐄^h)𝐕h×𝐕h×𝐌h such that

(𝐢κ𝐐h,𝐫)𝒯h+(𝐄h,curl𝐫)𝒯h-𝐄^h,𝐧×𝐫𝒯h=0for all 𝐫𝐕h,(2.2)-(𝐢κ𝐄h,𝐯)𝒯h+(𝐐h,curl𝐯)𝒯h+𝐧×𝐐^h,𝐯T𝒯h=(𝐟,𝐯)𝒯hfor all 𝐯𝐕h,(2.3)𝐧×𝐐^h-λ𝐄^h,𝝁Ω=𝐠,𝝁Ωfor all 𝝁𝐌h,(2.4)𝐧×𝐐^h,𝝁𝒯h\Ω=0for all 𝝁𝐌h.(2.5)

Here the numerical flux 𝐐^h is defined as

𝐧×𝐐^h=𝐧×𝐐h+τh((𝐄h)T-𝐄^h)on 𝒯h,(2.6)

and the constant τh 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 τh=1κh.

As already alluded in Section 1, one advantage of the above HDG method is that both 𝐐h and 𝐄h can be eliminated from the scheme. It then leads to a system for 𝐄^h only. We refer the reader to [18] for the detailed explanation.

We conclude this section by recalling the approximation properties of the 𝐋2 projection operators

𝚷h:𝐋2(Ω)𝐕h,𝐏M:𝐋2(h)𝐌h,

which are defined by

(𝚷h𝐄,𝐯)𝒯h=(𝐄,𝐯)𝒯hfor all 𝐯𝐕h,(2.7)(𝐏M𝐮,𝝁)h=(𝐮,𝝁)hfor all 𝝁𝐌h.(2.8)

It is well known that these two operators satisfy the following estimates for s=1,2 (see [6, 14]):

𝐄-𝚷h𝐄0,Ωhs|𝐄|s,Ω,(2.9)𝐄-𝚷h𝐄0,𝒯hhs-12|𝐄|s,Ω,(2.10)𝐮-𝐏M𝐮0,𝒯hhs-12|𝐮|s,Ω.(2.11)

The above approximation properties of 𝚷h and 𝐏M 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 κ,h>0. First, we quote some regularity results for the solution 𝐄 to problem (1.1)–(1.2), their proofs can be found in [10, 8].

Suppose that div𝐟~=0 and

𝐠~𝐇T12(Ω):={𝐠~(H12(Ω))3:𝐠~𝐧=0 on Ω}.

Then the solution 𝐄 is in 𝐇δ(curl,Ω) for 12<δ1, and the estimate 𝐄𝐇δ(curl,Ω)κM(𝐟~,𝐠~) holds, where

𝐇δ(curl,Ω):={𝐮𝐇δ(Ω):curl𝐮𝐇δ(Ω)},𝐮𝐇δ(curl,Ω)2:=𝐮𝐇δ(Ω)2+curl𝐮𝐇δ(Ω)2.

In this paper, we shall assume that 𝐄𝐇2(Ω). On the other hand, we note that the solution 𝐄 of problem (1.1)–(1.2) may not belong to 𝐇2(Ω) in some cases. However, since (cf. [9])

EiH(div,Ω)H(curl,Ω)𝐇loc1(Ω),i=1,2,3,

where E1,E2,E3 are the components of the vector function 𝐄, it can be shown that 𝐄𝐇loc2(Ω). Furthermore, if 𝐄𝐇2(Ω), it can also be shown that (cf. [8, Remark 2.3])

𝐄𝐇2(Ω)κM(𝐟~,𝐠~).

Second, to derive our stability result, we need the following lemma.

Suppose (𝐐h,𝐄h,𝐄^h)𝐕h×𝐕h×𝐌h solves (2.2)–(2.5). Then there hold

τh(𝐄h)T-𝐄^h0,𝒯h2+𝐄^h0,Ω2𝐟0,Ω𝐄h0,Ω+𝐠0,Ω2,(3.1)κ𝐐h0,Ω2κ𝐄h0,Ω2+𝐟0,Ω2+𝐠0,Ω2,(3.2)-𝐢κ𝐄h+curl𝐐h0,Ω𝐟0,Ω+τhh-12(𝐄h)T-𝐄^h0,𝒯h,(3.3)𝐢κ𝐐h+curl𝐄h0,Ωh-12(𝐄h)T-𝐄^h0,𝒯h.(3.4)

Proof.

Setting 𝐫=𝐐h, 𝐯=𝐄h, 𝝁=𝐄^h in (2.2)–(2.5), we get

𝐢κ𝐐h0,Ω2+(𝐄h,curl𝐐h)𝒯h-𝐄^h,𝐧×𝐐h𝒯h=0,(3.5)-𝐢κ𝐄h0,Ω2+(curl𝐐h,𝐄h)𝒯h+τh(𝐄h)T-𝐄^h,(𝐄h)T𝒯h=(𝐟,𝐄h)𝒯h,(3.6)𝐧×𝐐^h,𝐄^h𝒯h=𝐠,𝐄^hΩ+λ𝐄^h0,Ω2.(3.7)

By (3.7), the complex conjugation of (3.5) can be rewritten as

-𝐢κ𝐐h0,Ω2+(curl𝐐h,𝐄h)𝒯h+τh(𝐄h)T-𝐄^h,𝐄^h𝒯h=𝐠,𝐄^hΩ+λ𝐄^h0,Ω2.(3.8)

Subtracting (3.8) from (3.6) yields

𝐢κ(𝐐h0,Ω2-𝐄h0,Ω2)+τh(𝐄h)T-𝐄^h0,𝒯h2+λ𝐄^h0,Ω2=(𝐟,𝐄h)𝒯h-𝐠,𝐄^hΩ,

which implies (3.1) and (3.2).

Using Green’s formula, equations (2.2) and (2.3) can be rewritten as

(𝐢κ𝐐h+curl𝐄h,𝐫)𝒯h+(𝐄h)T-𝐄^h,𝐧×𝐫𝒯h=0for all 𝐫𝐕h,(3.9)(-𝐢κ𝐄h+curl𝐐h,𝐯)𝒯h+τh(𝐄h)T-𝐄^h,𝐯T𝒯h=(𝐟,𝐯)𝒯hfor all 𝐯𝐕h.(3.10)

Taking 𝐫=𝐢κ𝐐h+curl𝐄h and 𝐯=-𝐢κ𝐄h+curl𝐐h in (3.9) and (3.10), respectively, and using the trace inequality (cf. [20])

z0,Th-12z0,Tfor all zP1(T),(3.11)

we obtain (3.3) and (3.4). ∎

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 𝐯h𝐕h, there exist 𝐯hc𝐇0(curl,Ω)𝐕h and 𝐯~hc𝐕hc:=𝐇(curl,Ω)𝐕h such that

𝐯h-𝐯hc0,Ω+hcurl(𝐯h-𝐯hc)0,Ωh12(Fh(𝐯h)TF2)12,𝐯h-𝐯~hc0,Ω+hcurl(𝐯h-𝐯~hc)0,Ωh12(FhI(𝐯h)TF2)12.

Let Uh0 be defined by

Uh0:={uH01(Ω):u|T𝒫2(T) for all T𝒯h}.

We introduce the following Helmholtz decomposition for 𝐯hc𝐇0(curl,Ω)𝐕h:

𝐯hc=𝐰h0+rh0=𝐇𝐰h0+Hrh0,

where rh0Uh0 and Hrh0H01(Ω) are defined by

(rh0,ϕh0)=(𝐯hc,ϕh0)for all ϕh0Uh0,(Hrh0,ϕ)=(𝐯hc,ϕ)for all ϕH01(Ω).

It is clear that the above decomposition is orthogonal in the 𝐋2-inner product and div𝐇𝐰h0=0. By [12, Lemma 4.4], we have

𝐰h0-𝐇𝐰h00,Ωh𝐇𝐰h01,Ωhcurl𝐇𝐰h00,Ω=hcurl𝐯hc0,Ω.

We are now ready to state our stability result.

Let (𝐐h,𝐄h,𝐄^h)𝐕h×𝐕h×𝐌h solve (2.2)–(2.5). Then

𝐄h0,Ω+𝐐h0,ΩCstab𝐟0,Ω+Cstab𝐠0,Ω,(3.12)

where Cstab:=(1+κ3h2+κ-32h-12).

Proof.

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 𝐄hc𝐇0(curl,Ω)𝐕h such that

𝐄h-𝐄hc0,Ω+hcurl(𝐄h-𝐄hc)0,Ωh12(Fh(𝐄h)TF2)12h12((𝐄h)T-𝐄^h0,𝒯h+𝐄^h0,Ω).(3.13)

The above 𝐄hc admits the Helmholtz decompositions

𝐄hc=𝐰h0+rh0=𝐇𝐰h0+Hrh0,

which satisfy

𝐰h0-𝐇𝐰h00,Ωhcurl𝐄hc0,Ωhcurl(𝐄h-𝐄hc)0,Ω+hcurl𝐄h0,Ωhcurl(𝐄h-𝐄hc)0,Ω+hcurl𝐄h+𝐢κ𝐐h0,Ω+κh𝐐h0,Ωh12((𝐄h)T-𝐄^h0,𝒯h+𝐄^h0,Ω)+κh𝐐h0,Ωh12A+κh𝐐h0,Ω,(3.14)

where we have used (3.13) and Lemma 3.3, and

A:=(1+κh)(𝐟0,Ω𝐄h0,Ω+𝐠0,Ω).

Consequently, (2.9), (2.10), (3.14) and (3.11) imply

(𝐰h0-𝐇𝐰h0)T0,𝒯h𝐇𝐰h0-Πh𝐇𝐰h00,𝒯h+𝐰h0-Πh𝐇𝐰h00,𝒯h𝐇𝐰h0-Πh𝐇𝐰h00,𝒯h+h-12(𝐇𝐰h0-Πh𝐇𝐰h00,Ω+𝐰h0-𝐇𝐰h00,Ω)h12𝐇𝐰h01,Ω+h-12𝐰h0-𝐇𝐰h00,Ωh12curl𝐄hc0,Ω+h-12𝐰h0-𝐇𝐰h00,ΩA+κh12𝐐h0,Ω.(3.15)

Step 2: Consider the auxiliary problem

𝐢κ𝚽+curl𝚿=0in Ω,(3.16)curl𝚽-𝐢κ𝚿=𝐇𝐰h0in Ω,(3.17)𝐧×𝚽+λ𝚿T=0on Ω,(3.18)

which has a unique solution (𝚽,𝚿) satisfying the estimate

𝚽0,Ω+κ-1𝚽1,Ω+κ-2𝚿2,Ω+𝚿0,Ω+𝚿T0,Ω𝐇𝐰h00,Ω.(3.19)

By (3.17), Green’s formula and the definition of the 𝐋2-projection, we obtain

(𝐄h,𝐇𝐰h0)𝒯h=(𝐄h,curl𝚽)𝒯h+𝐢κ(𝐄h,𝚿)𝒯h=(curl𝐄h,𝚽)𝒯h+(𝐄h)T,𝐧×𝚽𝒯h+𝐢κ(𝐄h,Πh𝚿)𝒯h=(𝐄h,curlΠh𝚽)𝒯h+(𝐄h)T,𝐧×(𝚽-Πh𝚽)𝒯h+𝐢κ(𝐄h,Πh𝚿)𝒯h.

Using (2.2), (2.3) and the fact that 𝐧×𝚽 is continuous across the inner faces, the above equality can be rewritten as

(𝐄h,𝐇𝐰h0)𝒯h=-𝐢κ(𝐐h,𝚽)𝒯h+(𝐄h)T-𝐄^h,𝐧×(𝚽-Πh𝚽)𝒯h+𝐄^h,𝐧×𝚽Ω+𝐢κ(𝐄h,Πh𝚿)𝒯h=-𝐢κ(𝐐h,𝚽)𝒯h+(𝐄h)T-𝐄^h,𝐧×(𝚽-Πh𝚽)𝒯h+𝐄^h,𝐧×𝚽Ω+(𝐐h,curlΠh𝚿)𝒯h+𝐧×𝐐^h,(Πh𝚿)T𝒯h-(𝐟,Πh𝚿)𝒯h.

On noting

(𝐐h,curlΠh𝚿)𝒯h=(𝐐h,curl𝚿)𝒯h+𝐧×𝐐h,(𝚿-Πh𝚿)T𝒯h,

and the fact that (2.4) implies

𝐧×𝐐^h,𝚿T𝒯h=PM𝐠,𝚿T𝒯h+λ𝐄^h,𝚿TΩ,

by (3.16) and (3.18), we get

(𝐄h,𝐇𝐰h0)𝒯h=τh(𝐄h)T-𝐄^h,(Πh𝚿-𝚿)T𝒯h+(𝐄h)T-𝐄^h,𝐧×(𝚽-Πh𝚽)𝒯h+PM𝐠,𝚿TΩ-(𝐟,Πh𝚿)𝒯h.

Since

(𝐄h,𝐇𝐰h0)𝒯h=(𝐄h-𝐄hc,𝐇𝐰h0)𝒯h+(𝐇𝐰h0,𝐇𝐰h0)𝒯h,

we obtain

𝐇𝐰h00,Ω2=(𝐄h,𝐇𝐰h0)𝒯h-(𝐄h-𝐄hc,𝐇𝐰h0)𝒯h=τh(𝐄h)T-𝐄^h,(Πh𝚿-𝚿)T𝒯h+(𝐄h)T-𝐄^h,𝐧×(𝚽-Πh𝚽)𝒯h+PM𝐠,𝚿TΩ-(𝐟,Πh𝚿)𝒯h-(𝐄h-𝐄hc,𝐇𝐰h0)𝒯hτh(𝐄h)T-𝐄^h0,𝒯hκ2h32𝐇𝐰h00,Ω+(𝐄h)T-𝐄^h0,𝒯hκh12𝐇𝐰h00,Ω+𝐠0,Ω𝐇𝐰h00,Ω+𝐄h-𝐄hc0,Ω𝐇𝐰h00,Ω+𝐟0,Ω𝐇𝐰h00,Ω,

where we have used (2.9), (2.10) and (3.19). By (3.13) and Lemma 3.3, we get

𝐇𝐰h00,Ω(κ32h+h)(𝐟0,Ω𝐄h0,Ω+𝐠0,Ω)+𝐟0,Ω+𝐠0,Ω.(3.20)

Next, we derive an upper bound for the term (𝐄h,𝐰h0-𝐇𝐰h0)𝒯h. Notice that

(𝐄h,𝐰h0-𝐇𝐰h0)𝒯h=(𝐄h-1𝐢κcurl𝐐h,𝐰h0-𝐇𝐰h0)𝒯h+1𝐢κ(curl𝐐h,𝐰h0-𝐇𝐰h0)𝒯h,

and

(curl𝐐h,𝐰h0-𝐇𝐰h0)𝒯h=𝐧×𝐐h,(𝐰h0-𝐇𝐰h0)T𝒯h.

Since 𝐧×𝐐^h is continuous across the inner faces, the above equality becomes

(curl𝐐h,𝐰h0-𝐇𝐰h0)𝒯h=-τh(𝐄h)T-𝐄^h,(𝐰h0-𝐇𝐰h0)T𝒯h,

which yields

(𝐄h,𝐰h0-𝐇𝐰h0)𝒯h=(𝐄h-1𝐢κcurl𝐐h,𝐰h0-𝐇𝐰h0)𝒯h+𝐢τhκ(𝐄h)T-𝐄^h,(𝐰h0-𝐇𝐰h0)T𝒯h.

Hence, it follows from Young’s inequality, (3.2), (3.14) and (3.15) that

|(𝐄h,𝐰h0-𝐇𝐰h0)𝒯h|𝐄h-1𝐢κcurl𝐐h0,Ω𝐰h0-𝐇𝐰h00,Ω+τhκ(𝐄h)T-𝐄^h0,𝒯h(𝐰h0-𝐇𝐰h0)T0,𝒯hA𝐄h-1𝐢κcurl𝐐h0,Ωh12+(κh𝐄h-1𝐢κcurl𝐐h0,Ω)2+Aτhκ(𝐄h)T-𝐄^h0,𝒯h+(τhκ(𝐄h)T-𝐄^h0,𝒯hκh12)2+δ𝐄h0,Ω2+δκ(𝐟0,Ω2+𝐠0,Ω2).(3.21)

Step 3: Setting 𝐯=rh0 in (2.3) yields (𝐄h,rh0)=0. Using Young’s inequality again, we get

𝐄h0,Ω2=(𝐄h,𝐄h-𝐄hc)𝒯h+(𝐄h,𝐰h0-𝐇𝐰h0)𝒯h+(𝐄h,𝐇𝐰h0)𝒯h𝐄h-𝐄hc0,Ω2+|(𝐄h,𝐰h0-𝐇𝐰h0)𝒯h|+𝐇𝐰h00,Ω2+δ𝐄h0,Ω2.

Choosing δ small enough and using (3.21), we obtain

𝐄h0,Ω2𝐄h-𝐄hc0,Ω2+𝐇𝐰h00,Ω2+1κ(𝐟0,Ω2+𝐠0,Ω2)+Ah12𝐄h-1𝐢κcurl𝐐h0,Ω+(κh𝐄h-1𝐢κcurl𝐐h0,Ω)2+Aτhκ(𝐄h)T-𝐄^h0,𝒯h+(τhh12(𝐄h)T-𝐄^h0,𝒯h)2.

Then (3.12) follows from Lemma 3.3, (3.13) and (3.20). ∎

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 (𝐑h,𝐔h,𝐔^h)𝐕h×𝐕h×𝐌h such that

𝐢κ(𝐐-𝐑h,𝐫)𝒯h+(𝐄-𝐔h,curl𝐫)𝒯h-𝐄T-𝐔^h,𝐧×𝐫𝒯h=0,(4.1)𝐢κ(𝐄-𝐔h,𝐯)𝒯h+(𝐐-𝐑h,curl𝐯)𝒯h+𝐧×(𝐐-𝐑^h),𝐯T𝒯h=0,(4.2)𝐧×(𝐐-𝐑^h)-λ(𝐄T-𝐔^h),𝝁Ω=0,(4.3)𝐧×𝐑^h,𝝁𝒯h\Ω=0,(4.4)

for all 𝐫,𝐯𝐕h and 𝝁𝐌h, where

𝐧×𝐑^h=𝐧×𝐑h+τh((𝐔h)T-𝐔^h)on 𝒯h.

It is easy to see that (𝐑h,𝐔h,𝐔^h) is an HDG elliptic projection of (𝐐,𝐄,𝐄T). Indeed we have the following result.

Scheme (4.1)–(4.4) has a unique solution (𝐑h,𝐔h,𝐔^h) and the following estimates hold:

𝐐-𝐑h0,ΩhM(𝐟~,𝐠~),(4.5)(𝐔h)T-𝐔^h0,𝒯hκh32M(𝐟~,𝐠~),(4.6)𝐄T-𝐔^h0,Ωκ12hM(𝐟~,𝐠~).(4.7)

Proof.

Since the problem is linear, it suffices to show (4.5)–(4.7). Setting 𝐫=Πh𝐐-𝐑h and 𝐯=Πh𝐄-𝐔h in (4.1) and (4.2), we get

𝐢κ𝐐-𝐑h0,Ω2+(Πh𝐄-𝐔h,curl(Πh𝐐-𝐑h))𝒯h-𝐄T-𝐔^h,𝐧×(Πh𝐐-𝐑h)𝒯h=𝐢κ𝐐-Πh𝐐0,Ω2,(4.8)𝐢κ𝐄-𝐔h0,Ω2+(curl(𝐐-𝐑h),Πh𝐄-𝐔h)-τh(𝐔h)T-𝐔^h,(Πh𝐄-𝐔h)T𝒯h=𝐢κ𝐄-Πh𝐄0,Ω2.(4.9)

Subtracting the complex conjugation of (4.8) from (4.9) and using Green’s formula on each tetrahedra yield

𝐢κ(𝐄-𝐔h0,Ω2+𝐐-𝐑h0,Ω2)+𝐧×(𝐐-Πh𝐐),(Πh𝐄-𝐔h)T𝒯h+𝐧×(Πh𝐐-𝐑h),𝐄T-𝐔^h𝒯h+τh(𝐔h)T-𝐔^h0,𝒯h2-τh(𝐔h)T-𝐔^h,(Πh𝐄)T-𝐔^h𝒯h=𝐢κ(𝐄-Πh𝐄0,Ω2+𝐐-Πh𝐐0,Ω2).(4.10)

Let

T1:=𝐧×(𝐐-Πh𝐐),(Πh𝐄-𝐔h)T𝒯h+𝐧×(Πh𝐐-𝐑h),𝐄T-𝐔^h𝒯h.

By the definition of 𝐏M, we have

𝐧×(Πh𝐐-𝐑h),𝐄T-𝐔^h𝒯h=𝐧×(Πh𝐐-𝐑h),𝐏M(𝐄T)-𝐔^h𝒯h.

Now setting 𝝁=PM(𝐄T)-𝐔^h in (4.3) and (4.4), we get

T1=𝐧×(𝐐-Πh𝐐),(Πh𝐄-𝐔h)T-𝐏M(𝐄T)+𝐔^h𝒯h+𝐧×(𝐐-𝐑h),𝐏M(𝐄T)-𝐔^h𝒯h=𝐧×(𝐐-Πh𝐐),(Πh𝐄-𝐔h)T-𝐏M(𝐄T)+𝐔^h𝒯h+𝐧×(𝐐-𝐑^h),𝐏M(𝐄T)-𝐔^h𝒯h+τh(𝐔h)T-𝐔^h,𝐏M(𝐄T)-𝐔^h𝒯h=𝐧×(𝐐-Πh𝐐),(Πh𝐄)T-𝐏M(𝐄T)𝒯h-𝐧×(𝐐-Πh𝐐),(𝐔h)T-𝐔^h𝒯h+λ𝐄T-𝐔^h,𝐏M(𝐄T)-𝐔^hΩ+τh(𝐔h)T-𝐔^h,𝐏M(𝐄T)-𝐔^h𝒯h.

Substituting the above expression into (4.10) yields

𝐢κ(𝐄-𝐔h0,Ω2+𝐐-𝐑h0,Ω2)+τh(𝐔h)T-𝐔^h0,𝒯h2+λ𝐄T-𝐔^h0,Ω2=-𝐧×(𝐐-Πh𝐐),(Πh𝐄)T-𝐏M(𝐄T)𝒯h+𝐧×(𝐐-Πh𝐐),(𝐔h)T-𝐔^h𝒯h-λ𝐄T-𝐔^h,𝐏M(𝐄T)-𝐄TΩ-τh(𝐔h)T-𝐔^h,𝐏M(𝐄T)-(Πh𝐄)T𝒯h   +𝐢κ(𝐄-Πh𝐄0,Ω2+𝐐-Πh𝐐0,Ω2).(4.11)

Taking the absolute value on both sides of (4.11) and using Young’s inequality, we obtain

κ(𝐄-𝐔h0,Ω2+𝐐-𝐑h0,Ω2)+τh(𝐔h)T-𝐔^h0,𝒯h2+λ𝐄T-𝐔^h0,Ω2κ(𝐄-Πh𝐄0,Ω2+𝐐-Πh𝐐0,Ω2)+τh-1𝐐-Πh𝐐0,𝒯h2   +𝐐-Πh𝐐0,𝒯h(Πh𝐄)T-𝐏M(𝐄T)0,𝒯h   +𝐏M(𝐄T)-𝐄T0,𝒯h2+τh𝐏M(𝐄T)-(Πh𝐄)T0,𝒯h2.

It follows from (2.9)–(2.11) that

κ(𝐄-𝐔h0,Ω2+𝐐-𝐑h0,Ω2)+τh(𝐔h)T-𝐔^h0,𝒯h2+λ𝐄T-𝐔^h0,Ω2κh2M2(𝐟~,𝐠~),

which implies (4.5)–(4.7). ∎

Next we use the duality argument to estimate the 𝐋2-norm of 𝐄-𝐔h.

Let (𝐑h,𝐔h,𝐔^h) be the solution of (4.1)–(4.4). Then there holds

𝐄-𝐔h0,Ωκh2M(𝐟~,𝐠~)+h12𝐄T-𝐔^h0,Ω.

Proof.

It follows from [15] that there exists Πhc𝐄𝐕hc such that

𝐄-Πhc𝐄m,Ωhs-m𝐄s,Ω,s=1,2,m=0,1,(4.12)𝐄-Πhc𝐄0,𝒯hhs-12𝐄s,Ω,s=1,2.(4.13)

Let 𝐙h:=Πhc𝐄-𝐔h. By Lemma 3.4, (4.6) and (4.13), there exists 𝐙hc𝐇0(curl,Ω)𝐕h such that

𝐙h-𝐙hc0,Ω+hcurl(𝐙h-𝐙hc)0,Ωh12(Fh(𝐙h)TF2)12h12((𝐄-Πhc𝐄)T0,Ω+𝐄T-𝐔^h0,Ω+(𝐔h)T-𝐔^h0,𝒯h)κh2M(𝐟~,𝐠~)+h12𝐄T-𝐔^h0,Ω.(4.14)

We introduce the Helmholtz decompositions for 𝐙hc:

𝐙hc=𝐰h0+rh0=𝐇𝐰h0+Hrh0with rh0Uh0,div𝐇𝐰h0=0.

Moreover, the decomposition is orthogonal in the 𝐋2-inner product and satisfies

𝐰h0-𝐇𝐰h00,Ωhcurl𝐙hc0,Ωhcurl(𝐙h-𝐙hc)0,Ω+hcurl𝐙h0,Ω.(4.15)

By (2.1) and Green’s formula, equation (4.1) can be rewritten as

(𝐢κ𝐑h+curl𝐔h,𝐫)𝒯h+(𝐔h)T-𝐔^h,𝐧×𝐫𝒯h=0for all 𝐫𝐕h.

Letting 𝐫=𝐢κ𝐑h+curl𝐔h in the above equality and using (4.6) and (3.11) yield

𝐢κ𝐑h+curl𝐔h0,Ωh-12(𝐔h)T-𝐔^h0,𝒯hκhM(𝐟~,𝐠~).(4.16)

Then by the triangle inequality, (2.1), (4.5), (4.12) and (4.16), we get

curl𝐙h0,Ωcurl(Πhc𝐄-𝐄)0,Ω+𝐢κ(𝐐-𝐑h)0,Ω+𝐢κ𝐑h+curl𝐔h0,ΩκhM(𝐟~,𝐠~).(4.17)

Substituting (4.14) and (4.17) into (4.15) yields

𝐰h0-𝐇𝐰h00,Ωκh2M(𝐟~,𝐠~)+h12𝐄T-𝐔^h0,Ω.(4.18)

Now we consider the following auxiliary problem:

𝐢κ𝚽+curl𝚿=0in Ω,(4.19)curl𝚽+𝐢κ𝚿=𝐇𝐰h0in Ω,(4.20)𝐧×𝚽+λ𝚿T=0on Ω.(4.21)

Similar to the proof in [10], it can be shown that there exists a unique (𝚽,𝚿) such that

κ-1𝚿2,Ω+𝚿1,Ω+𝚽1,Ω𝐇𝐰h00,Ω.(4.22)

Setting 𝐫=Πh𝚽 in (4.1) and using (4.20), we have

(𝐄-𝐔h,𝐇𝐰h0)𝒯h=(𝐄-𝐔h,curl𝚽)𝒯h-𝐢κ(𝐄-𝐔h,𝚿)𝒯h=(𝐄-𝐔h,curl(𝚽-Πh𝚽))𝒯h-𝐢κ(𝐐-𝐑h,Πh𝚽)𝒯h+𝐄T-𝐔^h,𝐧×Πh𝚽𝒯h-𝐢κ(𝐄-𝐔h,𝚿)𝒯h.

By Green’s formula and (4.19) we get

𝐢κ(𝐐-𝐑h,𝚽)𝒯h=(𝐐-𝐑h,curl𝚿)𝒯h=(curl(𝐐-𝐑h),𝚿)𝒯h-𝐧×(𝐐-𝐑h),𝚿T𝒯h.

Thus, taking 𝐯=Πh𝚿 in (4.2) and using Green’s formula yield

(𝐄-𝐔h,𝐇𝐰h0)𝒯h=A1+A2,

where

A1:=(curl(𝐄-𝐔h),𝚽-Πh𝚽)𝒯h+𝐢κ(𝐐-𝐑h,𝚽-Πh𝚽)𝒯h+(curl(𝐐-𝐑h),Πh𝚿-𝚿)𝒯h-𝐢κ(𝐄-𝐔h,𝚿-Πh𝚿)𝒯h,A2:=𝐄T-(𝐔h)T,𝐧×(𝚽-Πh𝚽)𝒯h+𝐧×(𝐐-𝐑h),𝚿T𝒯h+𝐄T-𝐔^h,𝐧×Πh𝚽𝒯h-τh(𝐔h)T-𝐔^h,(Πh𝚿)T𝒯h.

From (4.3) and (4.4), we have

𝐧×(𝐐-𝐑^h),𝐏M(𝚿T)𝒯h=λ𝐄T-𝐔^h,𝐏M(𝚿T)Ω.

Then by (4.21) and the fact that 𝐧×𝚽 is continuous across the inner faces, we get

A2=(𝐔h)T-𝐔^h,𝐧×(Πh𝚽-𝚽)𝒯h+𝐧×(𝐐-𝐑h),𝚿T-𝐏M(𝚿T)𝒯h+τh(𝐔h)T-𝐔^h,𝐏M(𝚿T)-(Πh𝚿)T𝒯h+λ𝐄T-𝐔^h,𝐏M(𝚿T)-𝚿TΩ.

Using the definition of 𝐏M, we can rewrite A2 as

A2=(𝐔h)T-𝐔^h,𝐧×(Πh𝚽-𝚽)𝒯h+𝐧×(𝐐-Πh𝐐),𝚿T-𝐏M(𝚿T)𝒯h+τh(𝐔h)T-𝐔^h,𝐏M(𝚿T)-(Πh𝚿)T𝒯h+λ𝐄T-(Πh𝐄)T,PM(𝚿T)-𝚿TΩ.

Hence, it follows from (4.6), (4.22) and the definition and approximation properties (2.10) and (2.11) of the 𝐋2-projection that

|(𝐄-𝐔h,𝐇𝐰h0)𝒯h|=|A1+A2|κh2M(𝐟~,𝐠~)𝐇𝐰h00,Ω.(4.23)

Since (rh0,𝐇𝐰h0)=0, we have

𝐇𝐰h00,Ω2=(𝐄-𝐔h,𝐇𝐰h0)𝒯h-(𝐄-Πhc𝐄,𝐇𝐰h0)𝒯h-(𝐙h-𝐙hc,𝐇𝐰h0)𝒯h-(𝐰h0-𝐇𝐰h0,𝐇𝐰h0)𝒯h.

Hence, (4.12), (4.23), (4.14) and (4.18) imply

𝐇𝐰h00,Ωκh2M(𝐟~,𝐠~)+h12𝐄T-𝐔^h0,Ω.(4.24)

From (4.2)–(4.4), we have (𝐄-𝐔h,rh0)𝒯h=0. Then

𝐄-𝐔h0,Ω2=(𝐄-𝐔h,𝐄-Πhc𝐄)𝒯h+(𝐄-𝐔h,𝐙h-𝐙hc)𝒯h+(𝐄-𝐔h,𝐰h0-𝐇𝐰h0)𝒯h+(𝐄-𝐔h,𝐇𝐰h0)𝒯h.

By (4.12), (4.14), (4.18) and (4.24), we get

𝐄-𝐔h0,Ω𝐄-Πh𝐄0,Ω+𝐙h-𝐙hc0,Ω+𝐰h0-𝐇𝐰h00,Ω+𝐇𝐰h00,Ωκh2M(𝐟~,𝐠~)+h12𝐄T-𝐔^h0,Ω,

which completes the proof. ∎

By Lemma 4.2 and (4.7), we easily get the following estimate:

𝐄-𝐔h0,Ω(κ12h32+κh2)M(𝐟~,𝐠~).(4.25)

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 𝐄-𝐔h0,Ω with the help of a better estimation for 𝐄T-𝐔^h0,Ω.

Define

Uh:={uH1(Ω):u|T𝒫2(T) for all T𝒯h},

where 𝒫2(T) denotes the set of all quadratic polynomials on T. Obviously, Uh is a subspace of 𝐕hc. The next lemma establishes both the Helmholtz decomposition and the discrete Helmholtz decomposition as well as their relationship for each 𝐯h𝐕hc.

For each 𝐯h𝐕hc, there exist two decompositions for 𝐯h:

𝐯h=r+𝐰=rh+𝐰h,

where rH1(Ω), rhUh, 𝐰𝐇1(Ω) and 𝐰h𝐕hc. Moreover, both decompositions are orthogonal in the 𝐋2-inner product, 𝐰 is divergence-free, and the following error estimate holds:

𝐰-𝐰h0,Ωhcurl𝐯h0,Ω.(4.26)

Proof.

From [2, Theorem 3.12], there exists a unique 𝐰𝐇(curl,Ω)𝐇(div,Ω) such that

curl𝐰=curl𝐯hin Ω,div𝐰=0in Ω,𝐰𝝂=0on Ω,

where

𝐇(div,Ω):={𝐮𝐋2(Ω):div𝐮𝐋2(Ω)}.

By [15, Remark 3.52] (also see [17, 4]), we obtain that 𝐰𝐇1(Ω) and

𝐰1,Ωcurl𝐰0,Ω=curl𝐯h0,Ω.(4.27)

Since curl(𝐯h-𝐰)=0, there exists rH1(Ω) such that 𝐯h-𝐰=r, hence 𝐯h=r+𝐰 and the orthogonality of r and 𝐰 in the 𝐋2-inner product is obvious.

Next, we define rhUh/𝒫0(Ω) such that

(rh,ψh)=(𝐯h,ψh)for all ψhUh/𝒫0(Ω).(4.28)

It is easy to check that rh is well defined. Let 𝐰h:=𝐯h-rh. Then (4.28) infers the orthogonality of rh and 𝐰h in the 𝐋2-inner product. By (4.28) and the fact that

(𝐯h,ψh)=(r,ψh)for all ψhUh/𝒫0(Ω),

we get

(𝐰-𝐰n,ψh)=0for all ψhUh/𝒫0(Ω).(4.29)

Let πN denote the interpolation operator to the second type Nédélec edge finite element space and πD 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.,

curl(πN𝐰)=πDcurl𝐰.

Since curl𝐰=curl𝐯h and curl𝐯h is piecewise constant, we get

curlπN𝐰=πDcurl𝐰=πDcurl𝐯h=curl𝐯h=curl𝐰h,

which implies that there exists ϕhUh/𝒫0(Ω) such that πN𝐰-𝐰h=ϕh. Hence using (4.29), we get

𝐰-𝐰h0,Ω2=(𝐰-𝐰h,𝐰-πN𝐰)+(𝐰-𝐰h,πN𝐰-𝐰h)=(𝐰-𝐰h,𝐰-πN𝐰)𝐰-𝐰h0,Ω𝐰-πN𝐰0,Ω.

Hence,

𝐰-𝐰h0,Ω𝐰-πN𝐰0,Ω.

Finally, it follows from [15, Theorem 8.15] and (4.27) that

𝐰-𝐰h0,Ω𝐰-πN𝐰0,Ωh(𝐰1,Ω+curl𝐰0,Ω)hcurl𝐰0,Ω=hcurl𝐯h0,Ω.

The proof is complete. ∎

We now derive an improved estimate for 𝐄T-𝐔^h0,Ω in the mesh regime κh1.

Let (𝐑h,𝐔h,𝐔^h) be the solution of (4.1)–(4.4). Then the following estimate holds for κh1:

𝐄T-𝐔^h0,Ωκh32M(𝐟~,𝐠~).

Proof.

Recall that 𝐙h=Πhc𝐄-𝐔h. By Lemma 3.4 and (4.6), there exists 𝐙~hc𝐕hc such that

𝐙h-𝐙~hc0,Ω+hcurl(𝐙h-𝐙~hc)0,Ωh12(FhI(𝐙h)TF2)12h12(𝐔h)T-𝐔^h0,𝒯hκh2M(𝐟~,𝐠~).(4.30)

Also, by Lemma 4.4, we have

𝐙~hc=r+𝐰=rh+𝐰h,(4.31)

where rH1(Ω), rhUh, 𝐰𝐇1(Ω), 𝐰h𝐕hc, and 𝐰 is divergence free. It follows from (4.26), (4.30) and (4.17) that

𝐰-𝐰h0,Ωhcurl𝐙~hc0,Ωhcurl(𝐙h-𝐙~hc)0,Ω+hcurl𝐙h0,Ωκh2M(𝐟~,𝐠~).(4.32)

Similar to the derivation of the estimate for 𝐇𝐰h00,Ω in Lemma 4.2, it can be shown that

𝐰0,Ωκh2M(𝐟~,𝐠~).(4.33)

Setting 𝐫=rh in (4.2) and using (4.3) and (4.4), we get

𝐢κ(𝐄-𝐔h,rh)𝒯h+λ𝐄T-𝐔^h,(rh)TΩ=0.(4.34)

Since

(rh)T=(𝐙~hc-𝐙h)T+(Πhc𝐄-𝐄)T+(𝐄T-𝐔^h)-((𝐔h)T-𝐔^h)-(𝐰h)T,

substituting the above equality into (4.34) yields

λ𝐄T-𝐔^h0,Ω2=λ𝐄T-𝐔^h,(𝐙h-𝐙~hc)TΩ-λ𝐄T-𝐔^h,(Πhc𝐄-𝐄)TΩ+λ𝐄T-𝐔^h,(𝐔h)T-𝐔^hΩ+λ𝐄T-𝐔^h,(𝐰h)TΩ-𝐢κ(𝐄-𝐔h,rh)𝒯h.

By Young’s inequality, we have

𝐄T-𝐔^h0,Ω2(𝐙h-𝐙~hc)T0,Ω2+(Πhc𝐄-𝐄)T0,Ω2+(𝐔h)T-𝐔^h0,Ω2+(𝐰h)T0,Ω2+κ𝐄-𝐔h0,Ωrh0,Ω.(4.35)

By (4.32), (4.33) and (3.11), we get

(𝐰h)T0,Ω21h𝐰h0,Ω21h(𝐰-𝐰h0,Ω2+𝐰0,Ω2)κ2h3M2(𝐟~,𝐠~).(4.36)

It follows from the orthogonality of the decomposition (4.31), the triangle inequality, (4.12), (4.25) and (4.30) that, for κh1,

rh0,Ω𝐙h-𝐙~hc0,Ω+𝐙h0,Ωκ12h32M(𝐟~,𝐠~).(4.37)

Finally, combining (4.6), (4.12), (4.25), (4.30), (4.36) and (4.37) with (4.35) and using (3.11), we obtain

𝐄T-𝐔^h0,Ωκh32M(𝐟~,𝐠~),

The proof is complete. ∎

Now we are ready to state our optimal error estimate for 𝐄-𝐔h0,Ω.

Let (𝐑h,𝐔h,𝐔^h) be the solution of (4.1)–(4.4). Then the following estimate holds for all κ,h>0:

𝐄-𝐔h0,Ωκh2M(𝐟~,𝐠~).(4.38)

Proof.

If the mesh size h satisfies κh1, then (4.38) follows immediately from (4.25). If κh1, then (4.38) is a consequence of Lemma 4.2 and Theorem 4.5. ∎

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.

Let (𝐐h,𝐄h,𝐄^h) be the solution of (2.2)–(2.5). Then there hold

𝐄-𝐄h0,Ω(κh2+Cstabκ2h2)M(𝐟~,𝐠~),(4.39)κ𝐐-𝐐h0,Ω(κh+Cstabκ3h2)M(𝐟~,𝐠~).(4.40)

Moreover, if κ3h21, then

𝐄h0,Ω+𝐐h0,Ω𝐟0,Ω+𝐠0,Ω,(4.41)𝐄-𝐄h0,Ω(κh2+κ2h2)M(𝐟~,𝐠~),(4.42)κ𝐐-𝐐h0,Ω(κh+κ3h2)M(𝐟~,𝐠~).(4.43)

Proof.

Let ϵh𝐐:=𝐐h-𝐑h, ϵh𝐄:=𝐄h-𝐔h, and ϵh𝐄^:=𝐄^h-𝐔^h, By (2.2)–(2.5) and (4.1)–(4.4), we have

(ϵh𝐐,ϵh𝐄,ϵh𝐄^)𝐕h×𝐕h×𝐌h

and

(𝐢κϵh𝐐,𝐫)𝒯h+(ϵh𝐄,curl𝐫)𝒯h-ϵh𝐄^,𝐧×𝐫𝒯h=0,-(𝐢κϵh𝐄,𝐯)𝒯h+(ϵh𝐐,curl𝐯)𝒯h+𝐧×ϵ^h𝐐,𝐯T𝒯h=-2𝐢κ(𝐄-𝐔h,𝐯)𝒯h,𝐧×ϵ^h𝐐-λϵh𝐄^,𝝁Ω=0,𝐧×ϵ^h𝐐,𝝁𝒯h\Ω=0

for all 𝐫,𝐯𝐕h and 𝝁𝐌h. Moreover, the numerical flux ϵ^h𝐐 satisfies

𝐧×ϵ^h𝐐=𝐧×ϵh𝐐+τh((ϵh𝐄)T-ϵh𝐄^)on 𝒯h.

Applying the stability estimate in Theorem 3.5 to the above formulation yields

ϵh𝐄0,Ω+ϵh𝐐0,ΩCstabκ𝐄-𝐔h0,Ω.

Finally, (4.39) and (4.40) follow from an application of the triangle inequality and Theorem 4.6, and (4.41)–(4.43) follow from the “stability-error iterative improvement” technique of [7]. ∎

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 τh=1κh. The implementation is done using the MATLAB software package iFEM (cf. [5]).

Test 1. The relative errors
∥𝐐-𝐐h∥0,Ω/∥𝐐∥0,Ω${\lVert\mathbf{Q}-\mathbf{Q}_{h}\rVert_{0,\Omega}/\lVert\mathbf{Q}\rVert_{0,%
\Omega}}$ and
∥𝐄-𝐄h∥0,Ω/∥𝐄∥0,Ω${\lVert\mathbf{E}-\mathbf{E}_{h}\rVert_{0,\Omega}/\lVert\mathbf{E}\rVert_{0,%
\Omega}}$ of the HDG solution
for κ=10,20,30${\kappa=10,20,30}$.Test 1. The relative errors
∥𝐐-𝐐h∥0,Ω/∥𝐐∥0,Ω${\lVert\mathbf{Q}-\mathbf{Q}_{h}\rVert_{0,\Omega}/\lVert\mathbf{Q}\rVert_{0,%
\Omega}}$ and
∥𝐄-𝐄h∥0,Ω/∥𝐄∥0,Ω${\lVert\mathbf{E}-\mathbf{E}_{h}\rVert_{0,\Omega}/\lVert\mathbf{E}\rVert_{0,%
\Omega}}$ of the HDG solution
for κ=10,20,30${\kappa=10,20,30}$.
Figure 1

Test 1. The relative errors 𝐐-𝐐h0,Ω/𝐐0,Ω and 𝐄-𝐄h0,Ω/𝐄0,Ω of the HDG solution for κ=10,20,30.

We consider the time-harmonic Maxwell problem (1.1)–(1.2) in the unit cube Ω=[0,1]×[0,1]×[0,1] with 𝐟~, and 𝐠 is chosen such that the exact solution is given by

𝐄=(e𝐢κz,e𝐢κx,e𝐢κy)T.

For fixed wave number κ, we first compute the relative errors 𝐐-𝐐h0,Ω/𝐐0,Ω and 𝐄-𝐄h0,Ω/𝐄0,Ω, respectively. The left graph in Figure 1 displays the relative errors 𝐐-𝐐h0,Ω/𝐐0,Ω of the HDG solution for κ=10,20,30, while the right graph shows the same errors for 𝐄-𝐄h0,Ω/𝐄0,Ω. 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.

Test 1. Left: The relative errors of the HDG solution under the mesh condition κ⁢h=2${\kappa h=2}$.
Right: The relative errors of the HDG solution under the mesh condition κ3⁢h2=2${\kappa^{3}h^{2}=2}$.Test 1. Left: The relative errors of the HDG solution under the mesh condition κ⁢h=2${\kappa h=2}$.
Right: The relative errors of the HDG solution under the mesh condition κ3⁢h2=2${\kappa^{3}h^{2}=2}$.
Figure 2

Test 1. Left: The relative errors of the HDG solution under the mesh condition κh=2. Right: The relative errors of the HDG solution under the mesh condition κ3h2=2.

Test 1. Left: The traces of the real part of the first component of the
HDG solution for κ=16${\kappa=16}$ and h=1/16${h=1/16}$.Right: The traces of the same quantity for
κ=32${\kappa=32}$ and h=1/32${h=1/32}$.Test 1. Left: The traces of the real part of the first component of the
HDG solution for κ=16${\kappa=16}$ and h=1/16${h=1/16}$.Right: The traces of the same quantity for
κ=32${\kappa=32}$ and h=1/32${h=1/32}$.
Figure 3

Test 1. Left: The traces of the real part of the first component of the HDG solution for κ=16 and h=1/16.Right: The traces of the same quantity for κ=32 and h=1/32.

We present the relative errors 𝐐-𝐐h0,Ω/𝐐0,Ω and 𝐄-𝐄h0,Ω/𝐄0,Ω 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 κh=2. We observe that the relative errors cannot be controlled by κh 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 κ3h2=2. 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 κ=16,32 and show the phase error of the HDG solutions. We focus on the line segment x=0.5,y=0.5,0z1 and observe the traces of the real part of the first component of the HDG solutions with mesh sizes h=1/16 and h=1/32, 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 κh=1.

Test 2. Left: The relative errors of the HDG solution under the mesh condition κ⁢h=2${\kappa h=2}$.
Right: The relative errors of the HDG solution under the mesh condition κ3⁢h2=2${\kappa^{3}h^{2}=2}$.Test 2. Left: The relative errors of the HDG solution under the mesh condition κ⁢h=2${\kappa h=2}$.
Right: The relative errors of the HDG solution under the mesh condition κ3⁢h2=2${\kappa^{3}h^{2}=2}$.
Figure 4

Test 2. Left: The relative errors of the HDG solution under the mesh condition κh=2. Right: The relative errors of the HDG solution under the mesh condition κ3h2=2.

We consider the time-harmonic Maxwell problem (1.1)–(1.2) in the unit cube Ω=[0,1]×[0,1]×[0,1] with 𝐟 and 𝐠 chosen such that the exact solution is given by

𝐄=(sin(κy)J0(κr),cos(κz)J0(κr),𝐢κJ0(κr))T

in the polar coordinates, where r=(x2+y2+z2)12 and J0(z) is the Bessel function of the first kind. Although we assume that the right-hand side 𝐟 of equation (1.1) is a solenoidal current density, we find that the numerical results are also valid for the case div𝐟0.

The left graph in Figure 4 displays the relative errors 𝐐-𝐐h0,Ω/𝐐0,Ω and 𝐄-𝐄h0,Ω/𝐄0,Ω when the mesh condition κh=2 is enforced. It shows that the relative errors can not be controlled by κh 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 κ3h2=2. It clearly indicates that the relative errors can be controlled under this mesh condition.

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About the article

Received: 2015-10-23

Revised: 2016-01-11

Accepted: 2016-03-01

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).


Citation Information: Computational Methods in Applied Mathematics, Volume 16, Issue 3, Pages 429–445, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2016-0021.

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