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

Xiaobing Feng; Peipei Lu; Xuejun Xu

doi:10.1515/cmam-2016-0021

Article Publicly Available

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

Xiaobing Feng , Peipei Lu and Xuejun Xu

Published/Copyright: June 1, 2016

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From the journal Computational Methods in Applied Mathematics Volume 16 Issue 3

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

(1.1)curl⁡curl⁡𝐄-κ2⁢𝐄=𝐟~ in ⁢Ω,

(1.2)curl⁡𝐄×𝐧-𝐢⁢κ⁢λ⁢𝐄T=𝐠~ on ⁢∂⁡Ω,

where Ω⊂ℝ3 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:

∥𝐄h∥0,Ω+∥𝐐h∥0,Ω≲Cstab⁢∥𝐟∥0,Ω+Cstab⁢∥𝐠∥0,∂⁡Ω,

∥𝐄-𝐄h∥0,Ω≲(κ⁢h2+Cstab⁢κ2⁢h2)⁢M⁢(𝐟~,𝐠~),

(1.3)κ⁢∥𝐐-𝐐h∥0,Ω≲(κ⁢h+Cstab⁢κ3⁢h2)⁢M⁢(𝐟~,𝐠~),

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

𝐐:=𝐢κ⁢curl⁡𝐄,𝐟:=𝐢κ⁢𝐟~,𝐠:=-𝐢κ⁢𝐠~,Cstab:=1+κ3⁢h2+κ-32⁢h-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 κ⁢h≲1. However, estimate (1.3) indicates that the errors are not completely controlled by the product κ⁢h and they grow in κ while κ⁢h 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 κ3⁢h2≲1:

∥𝐄h∥0,Ω+∥𝐐h∥0,Ω≲∥𝐟∥0,Ω+∥𝐠∥0,∂⁡Ω,

∥𝐄-𝐄h∥0,Ω≲(κ⁢h2+κ2⁢h2)⁢M⁢(𝐟~,𝐠~),

κ⁢∥𝐐-𝐐h∥0,Ω≲(κ⁢h+κ3⁢h2)⁢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 A≲B and A≳B for the inequalities A≤C⁢B and A≥C⁢B, 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 λ≃1 and κ≥1.

Let Ω⊂ℝ3 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:=maxT∈𝒯h⁡hT. The set of all faces of 𝒯h is denoted by ℱh, the set of all interior faces by ℱhI, and the set of all boundary faces by ℱhB. We also define the jump 〚𝐯〛 of 𝐯 on an interior face ℱ=∂⁡T+∩∂⁡T- as

〚𝐯〛|ℱ:={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:={𝐯∈𝐋2(Ω):𝐯|T∈(P1(T))3 for all T∈𝒯h},

𝐌h:={𝝁∈𝐋2(ℱh):𝝁|F∈(P1(F))3 for all F∈ℱh},

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

(𝐯,𝐰)𝒯h:=∑T∈𝒯h(𝐯,𝐰)T ⁢for all ⁢𝐯,𝐰∈𝐋2⁢(Ω),

〈𝐯,𝐰〉∂⁡𝒯h:=∑T∈𝒯h〈𝐯,𝐰〉∂⁡T ⁢for all ⁢𝐯,𝐰∈𝐋2⁢(ℱh).

Here

(𝐯,𝐰)T:=∫T𝐯⋅𝐰¯⁢𝑑x and 〈𝐯,𝐰〉∂⁡T:=∫∂⁡T𝐯⋅𝐰¯⁢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

(2.1)𝐢⁢κ⁢𝐐+curl⁡𝐄=0 ⁢in ⁢Ω,

curl⁡𝐐-𝐢⁢κ⁢𝐄=𝐟 ⁢in ⁢Ω,

𝐧×𝐐-λ⁢𝐄T=𝐠 ⁢on ⁢∂⁡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

(2.2)(𝐢⁢κ⁢𝐐h,𝐫)𝒯h+(𝐄h,curl⁡𝐫)𝒯h-〈𝐄^h,𝐧×𝐫〉∂⁡𝒯h=0 ⁢for all ⁢𝐫∈𝐕h,

(2.3)-(𝐢⁢κ⁢𝐄h,𝐯)𝒯h+(𝐐h,curl⁡𝐯)𝒯h+〈𝐧×𝐐^h,𝐯T〉∂⁡𝒯h=(𝐟,𝐯)𝒯h ⁢for all ⁢𝐯∈𝐕h,

(2.4)〈𝐧×𝐐^h-λ⁢𝐄^h,𝝁〉∂⁡Ω=〈𝐠,𝝁〉∂⁡Ω ⁢for all ⁢𝝁∈𝐌h,

(2.5)〈𝐧×𝐐^h,𝝁〉∂⁡𝒯h\∂⁡Ω=0 ⁢for all ⁢𝝁∈𝐌h.

Here the numerical flux 𝐐^h is defined as

(2.6)𝐧×𝐐^h=𝐧×𝐐h+τh⁢((𝐄h)T-𝐄^h) on ⁢∂⁡𝒯h,

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=1κ⁢h.

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⁢(ℱh)→𝐌h,

which are defined by

(2.7)(𝚷h⁢𝐄,𝐯)𝒯h=(𝐄,𝐯)𝒯h ⁢for all ⁢𝐯∈𝐕h,

(2.8)(𝐏M⁢𝐮,𝝁)ℱh=(𝐮,𝝁)ℱh ⁢for all ⁢𝝁∈𝐌h.

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

(2.9)∥𝐄-𝚷h⁢𝐄∥0,Ω≲hs⁢|𝐄|s,Ω,

(2.10)∥𝐄-𝚷h⁢𝐄∥0,∂⁡𝒯h≲hs-12⁢|𝐄|s,Ω,

(2.11)∥𝐮-𝐏M⁢𝐮∥0,∂⁡𝒯h≲hs-12⁢|𝐮|s,Ω.

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].

Lemma 3.1

Suppose that div⁡𝐟~=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<δ≤1, and the estimate ∥𝐄∥𝐇δ⁢(curl,Ω)≲κ⁢M⁢(𝐟~,𝐠~) holds, where

𝐇δ⁢(curl,Ω):={𝐮∈𝐇δ⁢(Ω):curl⁡𝐮∈𝐇δ⁢(Ω)},∥𝐮∥𝐇δ⁢(curl,Ω)2:=∥𝐮∥𝐇δ⁢(Ω)2+∥curl⁡𝐮∥𝐇δ⁢(Ω)2.

Remark 3.2

In this paper, we shall assume that 𝐄∈𝐇2⁢(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])

∇⁡Ei∈H⁢(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 𝐄∈𝐇loc2⁢(Ω). Furthermore, if 𝐄∈𝐇2⁢(Ω), it can also be shown that (cf. [8, Remark 2.3])

∥𝐄∥𝐇2⁢(Ω)≲κ⁢M⁢(𝐟~,𝐠~).

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

Lemma 3.3

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

(3.1)τh⁢∥(𝐄h)T-𝐄^h∥0,∂⁡𝒯h2+∥𝐄^h∥0,∂⁡Ω2≲∥𝐟∥0,Ω⁢∥𝐄h∥0,Ω+∥𝐠∥0,∂⁡Ω2,

(3.2)κ⁢∥𝐐h∥0,Ω2≲κ⁢∥𝐄h∥0,Ω2+∥𝐟∥0,Ω2+∥𝐠∥0,∂⁡Ω2,

(3.3)∥-𝐢⁢κ⁢𝐄h+curl⁡𝐐h∥0,Ω≲∥𝐟∥0,Ω+τh⁢h-12⁢∥(𝐄h)T-𝐄^h∥0,∂⁡𝒯h,

(3.4)∥𝐢⁢κ⁢𝐐h+curl⁡𝐄h∥0,Ω≲h-12⁢∥(𝐄h)T-𝐄^h∥0,∂⁡𝒯h.

Proof.

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

(3.5)𝐢⁢κ⁢∥𝐐h∥0,Ω2+(𝐄h,curl⁡𝐐h)𝒯h-〈𝐄^h,𝐧×𝐐h〉∂⁡𝒯h=0,

(3.6)-𝐢⁢κ⁢∥𝐄h∥0,Ω2+(curl⁡𝐐h,𝐄h)𝒯h+τh⁢〈(𝐄h)T-𝐄^h,(𝐄h)T〉∂⁡𝒯h=(𝐟,𝐄h)𝒯h,

(3.7)〈𝐧×𝐐^h,𝐄^h〉∂⁡𝒯h=〈𝐠,𝐄^h〉∂⁡Ω+λ⁢∥𝐄^h∥0,∂⁡Ω2.

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

(3.8)-𝐢⁢κ⁢∥𝐐h∥0,Ω2+(curl⁡𝐐h,𝐄h)𝒯h+τh⁢〈(𝐄h)T-𝐄^h,𝐄^h〉∂⁡𝒯h=〈𝐠,𝐄^h〉∂⁡Ω+λ⁢∥𝐄^h∥0,∂⁡Ω2.

Subtracting (3.8) from (3.6) yields

𝐢⁢κ⁢(∥𝐐h∥0,Ω2-∥𝐄h∥0,Ω2)+τh⁢∥(𝐄h)T-𝐄^h∥0,∂⁡𝒯h2+λ⁢∥𝐄^h∥0,∂⁡Ω2=(𝐟,𝐄h)𝒯h-〈𝐠,𝐄^h〉∂⁡Ω,

which implies (3.1) and (3.2).

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

(3.9)(𝐢⁢κ⁢𝐐h+curl⁡𝐄h,𝐫)𝒯h+〈(𝐄h)T-𝐄^h,𝐧×𝐫〉∂⁡𝒯h=0 ⁢for all ⁢𝐫∈𝐕h,

(3.10)(-𝐢⁢κ⁢𝐄h+curl⁡𝐐h,𝐯)𝒯h+τh⁢〈(𝐄h)T-𝐄^h,𝐯T〉∂⁡𝒯h=(𝐟,𝐯)𝒯h ⁢for all ⁢𝐯∈𝐕h.

Taking 𝐫=𝐢⁢κ⁢𝐐h+curl⁡𝐄h and 𝐯=-𝐢⁢κ⁢𝐄h+curl⁡𝐐h in (3.9) and (3.10), respectively, and using the trace inequality (cf. [20])

(3.11)∥z∥0,∂⁡T≲h-12⁢∥z∥0,T for all ⁢z∈P1⁢(T),

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.

Lemma 3.4

For any 𝐯h∈𝐕h, there exist 𝐯hc∈𝐇0⁢(curl,Ω)∩𝐕h and 𝐯~hc∈𝐕hc:=𝐇⁢(curl,Ω)∩𝐕h such that

∥𝐯h-𝐯hc∥0,Ω+h∥curl(𝐯h-𝐯hc)∥0,Ω≲h12(∑F∈ℱh∥〚(𝐯h)T〛∥F2)12,

∥𝐯h-𝐯~hc∥0,Ω+h∥curl(𝐯h-𝐯~hc)∥0,Ω≲h12(∑F∈ℱhI∥〚(𝐯h)T〛∥F2)12.

Let Uh0 be defined by

Uh0:={u∈H01⁢(Ω):u|T∈𝒫2⁢(T)⁢ for all ⁢T∈𝒯h}.

We introduce the following Helmholtz decomposition for 𝐯hc∈𝐇0⁢(curl,Ω)∩𝐕h:

𝐯hc=𝐰h0+∇⁡rh0=𝐇𝐰h0+∇⁡H⁢rh0,

where rh0∈Uh0 and H⁢rh0∈H01⁢(Ω) are defined by

(∇⁡rh0,∇⁡ϕh0)=(𝐯hc,∇⁡ϕh0) ⁢for all ⁢ϕh0∈Uh0,

(∇⁡H⁢rh0,∇⁡ϕ)=(𝐯hc,∇⁡ϕ) ⁢for all ⁢ϕ∈H01⁢(Ω).

It is clear that the above decomposition is orthogonal in the 𝐋2-inner product and div⁡𝐇𝐰h0=0. By [12, Lemma 4.4], we have

∥𝐰h0-𝐇𝐰h0∥0,Ω≲h⁢∥𝐇𝐰h0∥1,Ω≲h⁢∥curl⁡𝐇𝐰h0∥0,Ω=h⁢∥curl⁡𝐯hc∥0,Ω.

We are now ready to state our stability result.

Theorem 3.5

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

(3.12)∥𝐄h∥0,Ω+∥𝐐h∥0,Ω≲Cstab⁢∥𝐟∥0,Ω+Cstab⁢∥𝐠∥0,∂⁡Ω,

where Cstab:=(1+κ3⁢h2+κ-32⁢h-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 𝐄hc∈𝐇0⁢(curl,Ω)∩𝐕h such that

(3.13)∥𝐄h-𝐄hc∥0,Ω+h∥curl(𝐄h-𝐄hc)∥0,Ω≲h12(∑F∈ℱh∥〚(𝐄h)T〛∥F2)12≲h12(∥(𝐄h)T-𝐄^h∥0,∂⁡𝒯h+∥𝐄^h∥0,∂⁡Ω).

The above 𝐄hc admits the Helmholtz decompositions

𝐄hc=𝐰h0+∇⁡rh0=𝐇𝐰h0+∇⁡H⁢rh0,

which satisfy

∥𝐰h0-𝐇𝐰h0∥0,Ω≲h⁢∥curl⁡𝐄hc∥0,Ω

≲h⁢∥curl⁡(𝐄h-𝐄hc)∥0,Ω+h⁢∥curl⁡𝐄h∥0,Ω

≲h⁢∥curl⁡(𝐄h-𝐄hc)∥0,Ω+h⁢∥curl⁡𝐄h+𝐢⁢κ⁢𝐐h∥0,Ω+κ⁢h⁢∥𝐐h∥0,Ω

≲h12⁢(∥(𝐄h)T-𝐄^h∥0,∂⁡𝒯h+∥𝐄^h∥0,∂⁡Ω)+κ⁢h⁢∥𝐐h∥0,Ω

(3.14)≲h12⁢A+κ⁢h⁢∥𝐐h∥0,Ω,

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

A:=(1+κ⁢h)⁢(∥𝐟∥0,Ω⁢∥𝐄h∥0,Ω+∥𝐠∥0,∂⁡Ω).

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

∥(𝐰h0-𝐇𝐰h0)T∥0,∂⁡𝒯h≲∥𝐇𝐰h0-Πh⁢𝐇𝐰h0∥0,∂⁡𝒯h+∥𝐰h0-Πh⁢𝐇𝐰h0∥0,∂⁡𝒯h

≲∥𝐇𝐰h0-Πh⁢𝐇𝐰h0∥0,∂⁡𝒯h+h-12⁢(∥𝐇𝐰h0-Πh⁢𝐇𝐰h0∥0,Ω+∥𝐰h0-𝐇𝐰h0∥0,Ω)

≲h12⁢∥𝐇𝐰h0∥1,Ω+h-12⁢∥𝐰h0-𝐇𝐰h0∥0,Ω

≲h12⁢∥curl⁡𝐄hc∥0,Ω+h-12⁢∥𝐰h0-𝐇𝐰h0∥0,Ω

(3.15)≲A+κ⁢h12⁢∥𝐐h∥0,Ω.

Step 2: Consider the auxiliary problem

(3.16)𝐢⁢κ⁢𝚽+curl⁡𝚿=0 ⁢in ⁢Ω,

(3.17)curl⁡𝚽-𝐢⁢κ⁢𝚿=𝐇𝐰h0 ⁢in ⁢Ω,

(3.18)𝐧×𝚽+λ⁢𝚿T=0 ⁢on ⁢∂⁡Ω,

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

(3.19)∥𝚽∥0,Ω+κ-1⁢∥𝚽∥1,Ω+κ-2⁢∥𝚿∥2,Ω+∥𝚿∥0,Ω+∥𝚿T∥0,∂⁡Ω≲∥𝐇𝐰h0∥0,Ω.

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

(𝐄h,𝐇𝐰h0)𝒯h=(𝐄h,curl⁡𝚽)𝒯h+𝐢⁢κ⁢(𝐄h,𝚿)𝒯h

=(curl⁡𝐄h,𝚽)𝒯h+〈(𝐄h)T,𝐧×𝚽〉∂⁡𝒯h+𝐢⁢κ⁢(𝐄h,Πh⁢𝚿)𝒯h

=(𝐄h,curl⁡Πh⁢𝚽)𝒯h+〈(𝐄h)T,𝐧×(𝚽-Πh⁢𝚽)〉∂⁡𝒯h+𝐢⁢κ⁢(𝐄h,Πh⁢𝚿)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=-𝐢⁢κ⁢(𝐐h,𝚽)𝒯h+〈(𝐄h)T-𝐄^h,𝐧×(𝚽-Πh⁢𝚽)〉∂⁡𝒯h+〈𝐄^h,𝐧×𝚽〉∂⁡Ω+𝐢⁢κ⁢(𝐄h,Πh⁢𝚿)𝒯h

=-𝐢⁢κ⁢(𝐐h,𝚽)𝒯h+〈(𝐄h)T-𝐄^h,𝐧×(𝚽-Πh⁢𝚽)〉∂⁡𝒯h+〈𝐄^h,𝐧×𝚽〉∂⁡Ω

+(𝐐h,curl⁡Πh⁢𝚿)𝒯h+〈𝐧×𝐐^h,(Πh⁢𝚿)T〉∂⁡𝒯h-(𝐟,Πh⁢𝚿)𝒯h.

On noting

(𝐐h,curl⁡Πh⁢𝚿)𝒯h=(𝐐h,curl⁡𝚿)𝒯h+〈𝐧×𝐐h,(𝚿-Πh⁢𝚿)T〉∂⁡𝒯h,

and the fact that (2.4) implies

〈𝐧×𝐐^h,𝚿T〉∂⁡𝒯h=〈PM⁢𝐠,𝚿T〉∂⁡𝒯h+λ⁢〈𝐄^h,𝚿T〉∂⁡Ω,

by (3.16) and (3.18), we get

(𝐄h,𝐇𝐰h0)𝒯h=τh⁢〈(𝐄h)T-𝐄^h,(Πh⁢𝚿-𝚿)T〉∂⁡𝒯h

+〈(𝐄h)T-𝐄^h,𝐧×(𝚽-Πh⁢𝚽)〉∂⁡𝒯h+〈PM⁢𝐠,𝚿T〉∂⁡Ω-(𝐟,Πh⁢𝚿)𝒯h.

Since

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

we obtain

∥𝐇𝐰h0∥0,Ω2=(𝐄h,𝐇𝐰h0)𝒯h-(𝐄h-𝐄hc,𝐇𝐰h0)𝒯h

=τh⁢〈(𝐄h)T-𝐄^h,(Πh⁢𝚿-𝚿)T〉∂⁡𝒯h+〈(𝐄h)T-𝐄^h,𝐧×(𝚽-Πh⁢𝚽)〉∂⁡𝒯h

+〈PM⁢𝐠,𝚿T〉∂⁡Ω-(𝐟,Πh⁢𝚿)𝒯h-(𝐄h-𝐄hc,𝐇𝐰h0)𝒯h

≲τh⁢∥(𝐄h)T-𝐄^h∥0,∂⁡𝒯h⁢κ2⁢h32⁢∥𝐇𝐰h0∥0,Ω+∥(𝐄h)T-𝐄^h∥0,∂⁡𝒯h⁢κ⁢h12⁢∥𝐇𝐰h0∥0,Ω

+∥𝐠∥0,∂⁡Ω⁢∥𝐇𝐰h0∥0,Ω+∥𝐄h-𝐄hc∥0,Ω⁢∥𝐇𝐰h0∥0,Ω+∥𝐟∥0,Ω⁢∥𝐇𝐰h0∥0,Ω,

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

(3.20)∥𝐇𝐰h0∥0,Ω≲(κ32⁢h+h)⁢(∥𝐟∥0,Ω⁢∥𝐄h∥0,Ω+∥𝐠∥0,∂⁡Ω)+∥𝐟∥0,Ω+∥𝐠∥0,∂⁡Ω.

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

(𝐄h,𝐰h0-𝐇𝐰h0)𝒯h=(𝐄h-1𝐢⁢κ⁢curl⁡𝐐h,𝐰h0-𝐇𝐰h0)𝒯h+1𝐢⁢κ⁢(curl⁡𝐐h,𝐰h0-𝐇𝐰h0)𝒯h,

and

(curl⁡𝐐h,𝐰h0-𝐇𝐰h0)𝒯h=〈𝐧×𝐐h,(𝐰h0-𝐇𝐰h0)T〉∂⁡𝒯h.

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

(curl⁡𝐐h,𝐰h0-𝐇𝐰h0)𝒯h=-τh⁢〈(𝐄h)T-𝐄^h,(𝐰h0-𝐇𝐰h0)T〉∂⁡𝒯h,

which yields

(𝐄h,𝐰h0-𝐇𝐰h0)𝒯h=(𝐄h-1𝐢⁢κ⁢curl⁡𝐐h,𝐰h0-𝐇𝐰h0)𝒯h+𝐢⁢τhκ⁢〈(𝐄h)T-𝐄^h,(𝐰h0-𝐇𝐰h0)T〉∂⁡𝒯h.

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

|(𝐄h,𝐰h0-𝐇𝐰h0)𝒯h|≤∥𝐄h-1𝐢⁢κ⁢curl⁡𝐐h∥0,Ω⁢∥𝐰h0-𝐇𝐰h0∥0,Ω

+τhκ⁢∥(𝐄h)T-𝐄^h∥0,∂⁡𝒯h⁢∥(𝐰h0-𝐇𝐰h0)T∥0,∂⁡𝒯h

≲A⁢∥𝐄h-1𝐢⁢κ⁢curl⁡𝐐h∥0,Ω⁢h12+(κ⁢h⁢∥𝐄h-1𝐢⁢κ⁢curl⁡𝐐h∥0,Ω)2

+A⁢τhκ⁢∥(𝐄h)T-𝐄^h∥0,∂⁡𝒯h+(τhκ⁢∥(𝐄h)T-𝐄^h∥0,∂⁡𝒯h⁢κ⁢h12)2

(3.21)+δ⁢∥𝐄h∥0,Ω2+δκ⁢(∥𝐟∥0,Ω2+∥𝐠∥0,∂⁡Ω2).

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

∥𝐄h∥0,Ω2=(𝐄h,𝐄h-𝐄hc)𝒯h+(𝐄h,𝐰h0-𝐇𝐰h0)𝒯h+(𝐄h,𝐇𝐰h0)𝒯h

≲∥𝐄h-𝐄hc∥0,Ω2+|(𝐄h,𝐰h0-𝐇𝐰h0)𝒯h|+∥𝐇𝐰h0∥0,Ω2+δ⁢∥𝐄h∥0,Ω2.

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

∥𝐄h∥0,Ω2≲∥𝐄h-𝐄hc∥0,Ω2+∥𝐇𝐰h0∥0,Ω2+1κ⁢(∥𝐟∥0,Ω2+∥𝐠∥0,∂⁡Ω2)

+A⁢h12⁢∥𝐄h-1𝐢⁢κ⁢curl⁡𝐐h∥0,Ω+(κ⁢h⁢∥𝐄h-1𝐢⁢κ⁢curl⁡𝐐h∥0,Ω)2

+A⁢τhκ⁢∥(𝐄h)T-𝐄^h∥0,∂⁡𝒯h+(τh⁢h12⁢∥(𝐄h)T-𝐄^h∥0,∂⁡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

(4.1)𝐢⁢κ⁢(𝐐-𝐑h,𝐫)𝒯h+(𝐄-𝐔h,curl⁡𝐫)𝒯h-〈𝐄T-𝐔^h,𝐧×𝐫〉∂⁡𝒯h=0,

(4.2)𝐢⁢κ⁢(𝐄-𝐔h,𝐯)𝒯h+(𝐐-𝐑h,curl⁡𝐯)𝒯h+〈𝐧×(𝐐-𝐑^h),𝐯T〉∂⁡𝒯h=0,

(4.3)〈𝐧×(𝐐-𝐑^h)-λ⁢(𝐄T-𝐔^h),𝝁〉∂⁡Ω=0,

(4.4)〈𝐧×𝐑^h,𝝁〉∂⁡𝒯h\∂⁡Ω=0,

for all 𝐫,𝐯∈𝐕h and 𝝁∈𝐌h, 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.

Theorem 4.1

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

(4.5)∥𝐐-𝐑h∥0,Ω≲h⁢M⁢(𝐟~,𝐠~),

(4.6)∥(𝐔h)T-𝐔^h∥0,∂⁡𝒯h≲κ⁢h32⁢M⁢(𝐟~,𝐠~),

(4.7)∥𝐄T-𝐔^h∥0,∂⁡Ω≲κ12⁢h⁢M⁢(𝐟~,𝐠~).

Proof.

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

(4.8)𝐢⁢κ⁢∥𝐐-𝐑h∥0,Ω2+(Πh⁢𝐄-𝐔h,curl⁡(Πh⁢𝐐-𝐑h))𝒯h-〈𝐄T-𝐔^h,𝐧×(Πh⁢𝐐-𝐑h)〉∂⁡𝒯h=𝐢⁢κ⁢∥𝐐-Πh⁢𝐐∥0,Ω2,

(4.9)𝐢⁢κ⁢∥𝐄-𝐔h∥0,Ω2+(curl⁡(𝐐-𝐑h),Πh⁢𝐄-𝐔h)-τh⁢〈(𝐔h)T-𝐔^h,(Πh⁢𝐄-𝐔h)T〉∂⁡𝒯h=𝐢⁢κ⁢∥𝐄-Πh⁢𝐄∥0,Ω2.

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

𝐢⁢κ⁢(∥𝐄-𝐔h∥0,Ω2+∥𝐐-𝐑h∥0,Ω2)+〈𝐧×(𝐐-Πh⁢𝐐),(Πh⁢𝐄-𝐔h)T〉∂⁡𝒯h

+〈𝐧×(Πh⁢𝐐-𝐑h),𝐄T-𝐔^h〉∂⁡𝒯h+τh⁢∥(𝐔h)T-𝐔^h∥0,∂⁡𝒯h2-τh⁢〈(𝐔h)T-𝐔^h,(Πh⁢𝐄)T-𝐔^h〉∂⁡𝒯h

(4.10)=𝐢⁢κ⁢(∥𝐄-Πh⁢𝐄∥0,Ω2+∥𝐐-Πh⁢𝐐∥0,Ω2).

Let

T1:=〈𝐧×(𝐐-Πh⁢𝐐),(Πh⁢𝐄-𝐔h)T〉∂⁡𝒯h+〈𝐧×(Πh⁢𝐐-𝐑h),𝐄T-𝐔^h〉∂⁡𝒯h.

By the definition of 𝐏M, we have

〈𝐧×(Πh⁢𝐐-𝐑h),𝐄T-𝐔^h〉∂⁡𝒯h=〈𝐧×(Πh⁢𝐐-𝐑h),𝐏M⁢(𝐄T)-𝐔^h〉∂⁡𝒯h.

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

T1=〈𝐧×(𝐐-Πh⁢𝐐),(Πh⁢𝐄-𝐔h)T-𝐏M⁢(𝐄T)+𝐔^h〉∂⁡𝒯h

+〈𝐧×(𝐐-𝐑h),𝐏M⁢(𝐄T)-𝐔^h〉∂⁡𝒯h

=〈𝐧×(𝐐-Πh⁢𝐐),(Πh⁢𝐄-𝐔h)T-𝐏M⁢(𝐄T)+𝐔^h〉∂⁡𝒯h

+〈𝐧×(𝐐-𝐑^h),𝐏M⁢(𝐄T)-𝐔^h〉∂⁡𝒯h+τh⁢〈(𝐔h)T-𝐔^h,𝐏M⁢(𝐄T)-𝐔^h〉∂⁡𝒯h

=〈𝐧×(𝐐-Πh⁢𝐐),(Πh⁢𝐄)T-𝐏M⁢(𝐄T)〉∂⁡𝒯h-〈𝐧×(𝐐-Πh⁢𝐐),(𝐔h)T-𝐔^h〉∂⁡𝒯h

+λ⁢〈𝐄T-𝐔^h,𝐏M⁢(𝐄T)-𝐔^h〉∂⁡Ω+τh⁢〈(𝐔h)T-𝐔^h,𝐏M⁢(𝐄T)-𝐔^h〉∂⁡𝒯h.

Substituting the above expression into (4.10) yields

𝐢⁢κ⁢(∥𝐄-𝐔h∥0,Ω2+∥𝐐-𝐑h∥0,Ω2)+τh⁢∥(𝐔h)T-𝐔^h∥0,∂⁡𝒯h2+λ⁢∥𝐄T-𝐔^h∥0,∂⁡Ω2

=-〈𝐧×(𝐐-Πh⁢𝐐),(Πh⁢𝐄)T-𝐏M⁢(𝐄T)〉∂⁡𝒯h+〈𝐧×(𝐐-Πh⁢𝐐),(𝐔h)T-𝐔^h〉∂⁡𝒯h

-λ⁢〈𝐄T-𝐔^h,𝐏M⁢(𝐄T)-𝐄T〉∂⁡Ω-τh⁢〈(𝐔h)T-𝐔^h,𝐏M⁢(𝐄T)-(Πh⁢𝐄)T〉∂⁡𝒯h

(4.11) +𝐢⁢κ⁢(∥𝐄-Πh⁢𝐄∥0,Ω2+∥𝐐-Πh⁢𝐐∥0,Ω2).

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

κ⁢(∥𝐄-𝐔h∥0,Ω2+∥𝐐-𝐑h∥0,Ω2)+τh⁢∥(𝐔h)T-𝐔^h∥0,∂⁡𝒯h2+λ⁢∥𝐄T-𝐔^h∥0,∂⁡Ω2

≲κ⁢(∥𝐄-Πh⁢𝐄∥0,Ω2+∥𝐐-Πh⁢𝐐∥0,Ω2)+τh-1⁢∥𝐐-Πh⁢𝐐∥0,∂⁡𝒯h2

+∥𝐐-Πh⁢𝐐∥0,∂⁡𝒯h⁢∥(Πh⁢𝐄)T-𝐏M⁢(𝐄T)∥0,∂⁡𝒯h

+∥𝐏M⁢(𝐄T)-𝐄T∥0,∂⁡𝒯h2+τh⁢∥𝐏M⁢(𝐄T)-(Πh⁢𝐄)T∥0,∂⁡𝒯h2.

It follows from (2.9)–(2.11) that

κ⁢(∥𝐄-𝐔h∥0,Ω2+∥𝐐-𝐑h∥0,Ω2)+τh⁢∥(𝐔h)T-𝐔^h∥0,∂⁡𝒯h2+λ⁢∥𝐄T-𝐔^h∥0,∂⁡Ω2≲κ⁢h2⁢M2⁢(𝐟~,𝐠~),

which implies (4.5)–(4.7). ∎

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

Lemma 4.2

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

∥𝐄-𝐔h∥0,Ω≲κ⁢h2⁢M⁢(𝐟~,𝐠~)+h12⁢∥𝐄T-𝐔^h∥0,∂⁡Ω.

Proof.

It follows from [15] that there exists Πhc⁢𝐄∈𝐕hc such that

(4.12)∥𝐄-Πhc⁢𝐄∥m,Ω≲hs-m⁢∥𝐄∥s,Ω,s=1,2,m=0,1,

(4.13)∥𝐄-Πhc⁢𝐄∥0,∂⁡𝒯h≲hs-12⁢∥𝐄∥s,Ω,s=1,2.

Let 𝐙h:=Πhc⁢𝐄-𝐔h. By Lemma 3.4, (4.6) and (4.13), there exists 𝐙hc∈𝐇0⁢(curl,Ω)∩𝐕h such that

∥𝐙h-𝐙hc∥0,Ω+h∥curl(𝐙h-𝐙hc)∥0,Ω≲h12(∑F∈ℱh∥〚(𝐙h)T〛∥F2)12

≲h12⁢(∥(𝐄-Πhc⁢𝐄)T∥0,∂⁡Ω+∥𝐄T-𝐔^h∥0,∂⁡Ω+∥(𝐔h)T-𝐔^h∥0,∂⁡𝒯h)

(4.14)≲κ⁢h2⁢M⁢(𝐟~,𝐠~)+h12⁢∥𝐄T-𝐔^h∥0,∂⁡Ω.

We introduce the Helmholtz decompositions for 𝐙hc:

𝐙hc=𝐰h0+∇⁡rh0=𝐇𝐰h0+∇⁡H⁢rh0 with ⁢rh0∈Uh0,div⁡𝐇𝐰h0=0.

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

(4.15)∥𝐰h0-𝐇𝐰h0∥0,Ω≲h⁢∥curl⁡𝐙hc∥0,Ω≤h⁢∥curl⁡(𝐙h-𝐙hc)∥0,Ω+h⁢∥curl⁡𝐙h∥0,Ω.

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

(𝐢⁢κ⁢𝐑h+curl⁡𝐔h,𝐫)𝒯h+〈(𝐔h)T-𝐔^h,𝐧×𝐫〉∂⁡𝒯h=0 for all ⁢𝐫∈𝐕h.

Letting 𝐫=𝐢⁢κ⁢𝐑h+curl⁡𝐔h in the above equality and using (4.6) and (3.11) yield

(4.16)∥𝐢⁢κ⁢𝐑h+curl⁡𝐔h∥0,Ω≲h-12⁢∥(𝐔h)T-𝐔^h∥0,∂⁡𝒯h≲κ⁢h⁢M⁢(𝐟~,𝐠~).

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

(4.17)∥curl⁡𝐙h∥0,Ω≤∥curl⁡(Πhc⁢𝐄-𝐄)∥0,Ω+∥𝐢⁢κ⁢(𝐐-𝐑h)∥0,Ω+∥𝐢⁢κ⁢𝐑h+curl⁡𝐔h∥0,Ω≲κ⁢h⁢M⁢(𝐟~,𝐠~).

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

(4.18)∥𝐰h0-𝐇𝐰h0∥0,Ω≲κ⁢h2⁢M⁢(𝐟~,𝐠~)+h12⁢∥𝐄T-𝐔^h∥0,∂⁡Ω.

Now we consider the following auxiliary problem:

(4.19)𝐢⁢κ⁢𝚽+curl⁡𝚿=0 ⁢in ⁢Ω,

(4.20)curl⁡𝚽+𝐢⁢κ⁢𝚿=𝐇𝐰h0 ⁢in ⁢Ω,

(4.21)𝐧×𝚽+λ⁢𝚿T=0 ⁢on ⁢∂⁡Ω.

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

(4.22)κ-1⁢∥𝚿∥2,Ω+∥𝚿∥1,Ω+∥𝚽∥1,Ω≲∥𝐇𝐰h0∥0,Ω.

Setting 𝐫=Πh⁢𝚽 in (4.1) and using (4.20), we have

(𝐄-𝐔h,𝐇𝐰h0)𝒯h=(𝐄-𝐔h,curl⁡𝚽)𝒯h-𝐢⁢κ⁢(𝐄-𝐔h,𝚿)𝒯h

=(𝐄-𝐔h,curl⁡(𝚽-Πh⁢𝚽))𝒯h-𝐢⁢κ⁢(𝐐-𝐑h,Πh⁢𝚽)𝒯h

+〈𝐄T-𝐔^h,𝐧×Πh⁢𝚽〉∂⁡𝒯h-𝐢⁢κ⁢(𝐄-𝐔h,𝚿)𝒯h.

By Green’s formula and (4.19) we get

𝐢⁢κ⁢(𝐐-𝐑h,𝚽)𝒯h=(𝐐-𝐑h,curl⁡𝚿)𝒯h

=(curl⁡(𝐐-𝐑h),𝚿)𝒯h-〈𝐧×(𝐐-𝐑h),𝚿T〉∂⁡𝒯h.

Thus, taking 𝐯=Πh⁢𝚿 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),Πh⁢𝚿-𝚿)𝒯h-𝐢⁢κ⁢(𝐄-𝐔h,𝚿-Πh⁢𝚿)𝒯h,

A2:=〈𝐄T-(𝐔h)T,𝐧×(𝚽-Πh𝚽)〉∂⁡𝒯h+〈𝐧×(𝐐-𝐑h),𝚿T〉∂⁡𝒯h

+〈𝐄T-𝐔^h,𝐧×Πh⁢𝚽〉∂⁡𝒯h-τh⁢〈(𝐔h)T-𝐔^h,(Πh⁢𝚿)T〉∂⁡𝒯h.

From (4.3) and (4.4), we have

〈𝐧×(𝐐-𝐑^h),𝐏M⁢(𝚿T)〉∂⁡𝒯h=λ⁢〈𝐄T-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×(Πh⁢𝚽-𝚽)〉∂⁡𝒯h+〈𝐧×(𝐐-𝐑h),𝚿T-𝐏M⁢(𝚿T)〉∂⁡𝒯h

+τh⁢〈(𝐔h)T-𝐔^h,𝐏M⁢(𝚿T)-(Πh⁢𝚿)T〉∂⁡𝒯h+λ⁢〈𝐄T-𝐔^h,𝐏M⁢(𝚿T)-𝚿T〉∂⁡Ω.

Using the definition of 𝐏M, we can rewrite A2 as

A2=〈(𝐔h)T-𝐔^h,𝐧×(Πh⁢𝚽-𝚽)〉∂⁡𝒯h+〈𝐧×(𝐐-Πh⁢𝐐),𝚿T-𝐏M⁢(𝚿T)〉∂⁡𝒯h

+τh⁢〈(𝐔h)T-𝐔^h,𝐏M⁢(𝚿T)-(Πh⁢𝚿)T〉∂⁡𝒯h+λ⁢〈𝐄T-(Πh⁢𝐄)T,PM⁢(𝚿T)-𝚿T〉∂⁡Ω.

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

(4.23)|(𝐄-𝐔h,𝐇𝐰h0)𝒯h|=|A1+A2|≲κ⁢h2⁢M⁢(𝐟~,𝐠~)⁢∥𝐇𝐰h0∥0,Ω.

Since (∇⁡rh0,𝐇𝐰h0)=0, we have

∥𝐇𝐰h0∥0,Ω2=(𝐄-𝐔h,𝐇𝐰h0)𝒯h-(𝐄-Πhc⁢𝐄,𝐇𝐰h0)𝒯h-(𝐙h-𝐙hc,𝐇𝐰h0)𝒯h-(𝐰h0-𝐇𝐰h0,𝐇𝐰h0)𝒯h.

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

(4.24)∥𝐇𝐰h0∥0,Ω≲κ⁢h2⁢M⁢(𝐟~,𝐠~)+h12⁢∥𝐄T-𝐔^h∥0,∂⁡Ω.

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

∥𝐄-𝐔h∥0,Ω2=(𝐄-𝐔h,𝐄-Πhc⁢𝐄)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-𝐔h∥0,Ω≲∥𝐄-Πh⁢𝐄∥0,Ω+∥𝐙h-𝐙hc∥0,Ω+∥𝐰h0-𝐇𝐰h0∥0,Ω+∥𝐇𝐰h0∥0,Ω

≲κ⁢h2⁢M⁢(𝐟~,𝐠~)+h12⁢∥𝐄T-𝐔^h∥0,∂⁡Ω,

which completes the proof. ∎

Remark 4.3

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

(4.25)∥𝐄-𝐔h∥0,Ω≲(κ12⁢h32+κ⁢h2)⁢M⁢(𝐟~,𝐠~).

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-𝐔h∥0,Ω with the help of a better estimation for ∥𝐄T-𝐔^h∥0,∂⁡Ω.

Define

Uh:={u∈H1⁢(Ω):u|T∈𝒫2⁢(T)⁢ for all ⁢T∈𝒯h},

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 𝐯h∈𝐕hc.

Lemma 4.4

For each 𝐯h∈𝐕hc, there exist two decompositions for 𝐯h:

𝐯h=∇⁡r+𝐰=∇⁡rh+𝐰h,

where r∈H1⁢(Ω), rh∈Uh, 𝐰∈𝐇1⁢(Ω) and 𝐰h∈𝐕hc. Moreover, both decompositions are orthogonal in the 𝐋2-inner product, 𝐰 is divergence-free, and the following error estimate holds:

(4.26)∥𝐰-𝐰h∥0,Ω≲h⁢∥curl⁡𝐯h∥0,Ω.

Proof.

From [2, Theorem 3.12], there exists a unique 𝐰∈𝐇⁢(curl,Ω)∩𝐇⁢(div,Ω) such that

curl⁡𝐰=curl⁡𝐯h ⁢in ⁢Ω,

div⁡𝐰=0 ⁢in ⁢Ω,

𝐰⋅𝝂=0 ⁢on ⁢∂⁡Ω,

where

𝐇⁢(div,Ω):={𝐮∈𝐋2⁢(Ω):div⁡𝐮∈𝐋2⁢(Ω)}.

By [15, Remark 3.52] (also see [17, 4]), we obtain that 𝐰∈𝐇1⁢(Ω) and

(4.27)∥𝐰∥1,Ω≲∥curl⁡𝐰∥0,Ω=∥curl⁡𝐯h∥0,Ω.

Since curl⁡(𝐯h-𝐰)=0, there exists r∈H1⁢(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 rh∈Uh/𝒫0⁢(Ω) such that

(4.28)(∇⁡rh,∇⁡ψh)=(𝐯h,∇⁡ψh) for all ⁢ψh∈Uh/𝒫0⁢(Ω).

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 ⁢ψh∈Uh/𝒫0⁢(Ω),

we get

(4.29)(𝐰-𝐰n,∇⁡ψh)=0 for all ⁢ψh∈Uh/𝒫0⁢(Ω).

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⁡(πN⁢𝐰)=πD⁢curl⁡𝐰.

Since curl⁡𝐰=curl⁡𝐯h and curl⁡𝐯h is piecewise constant, we get

curl⁡πN⁢𝐰=πD⁢curl⁡𝐰=πD⁢curl⁡𝐯h=curl⁡𝐯h=curl⁡𝐰h,

which implies that there exists ϕh∈Uh/𝒫0⁢(Ω) such that πN⁢𝐰-𝐰h=∇⁡ϕh. Hence using (4.29), we get

∥𝐰-𝐰h∥0,Ω2=(𝐰-𝐰h,𝐰-πN⁢𝐰)+(𝐰-𝐰h,πN⁢𝐰-𝐰h)

=(𝐰-𝐰h,𝐰-πN⁢𝐰)≤∥𝐰-𝐰h∥0,Ω⁢∥𝐰-πN⁢𝐰∥0,Ω.

Hence,

∥𝐰-𝐰h∥0,Ω≤∥𝐰-πN⁢𝐰∥0,Ω.

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

∥𝐰-𝐰h∥0,Ω≤∥𝐰-πN⁢𝐰∥0,Ω≲h⁢(∥𝐰∥1,Ω+∥curl⁡𝐰∥0,Ω)

≲h⁢∥curl⁡𝐰∥0,Ω=h⁢∥curl⁡𝐯h∥0,Ω.

The proof is complete. ∎

We now derive an improved estimate for ∥𝐄T-𝐔^h∥0,∂⁡Ω in the mesh regime κ⁢h≤1.

Theorem 4.5

Let (𝐑h,𝐔h,𝐔^h) be the solution of (4.1)–(4.4). Then the following estimate holds for κ⁢h≤1:

∥𝐄T-𝐔^h∥0,∂⁡Ω≲κ⁢h32⁢M⁢(𝐟~,𝐠~).

Proof.

Recall that 𝐙h=Πhc⁢𝐄-𝐔h. By Lemma 3.4 and (4.6), there exists 𝐙~hc∈𝐕hc such that

(4.30)∥𝐙h-𝐙~hc∥0,Ω+h∥curl(𝐙h-𝐙~hc)∥0,Ω≲h12(∑F∈ℱhI∥〚(𝐙h)T〛∥F2)12≲h12∥(𝐔h)T-𝐔^h∥0,∂⁡𝒯h≲κh2M(𝐟~,𝐠~).

Also, by Lemma 4.4, we have

(4.31)𝐙~hc=∇⁡r+𝐰=∇⁡rh+𝐰h,

where r∈H1⁢(Ω), rh∈Uh, 𝐰∈𝐇1⁢(Ω), 𝐰h∈𝐕hc, and 𝐰 is divergence free. It follows from (4.26), (4.30) and (4.17) that

(4.32)∥𝐰-𝐰h∥0,Ω≲h⁢∥curl⁡𝐙~hc∥0,Ω≲h⁢∥curl⁡(𝐙h-𝐙~hc)∥0,Ω+h⁢∥curl⁡𝐙h∥0,Ω≲κ⁢h2⁢M⁢(𝐟~,𝐠~).

Similar to the derivation of the estimate for ∥𝐇𝐰h0∥0,Ω in Lemma 4.2, it can be shown that

(4.33)∥𝐰∥0,Ω≲κ⁢h2⁢M⁢(𝐟~,𝐠~).

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

(4.34)𝐢⁢κ⁢(𝐄-𝐔h,∇⁡rh)𝒯h+λ⁢〈𝐄T-𝐔^h,(∇⁡rh)T〉∂⁡Ω=0.

Since

(∇⁡rh)T=(𝐙~hc-𝐙h)T+(Πhc⁢𝐄-𝐄)T+(𝐄T-𝐔^h)-((𝐔h)T-𝐔^h)-(𝐰h)T,

substituting the above equality into (4.34) yields

λ⁢∥𝐄T-𝐔^h∥0,∂⁡Ω2=λ⁢〈𝐄T-𝐔^h,(𝐙h-𝐙~hc)T〉∂⁡Ω-λ⁢〈𝐄T-𝐔^h,(Πhc⁢𝐄-𝐄)T〉∂⁡Ω

+λ⁢〈𝐄T-𝐔^h,(𝐔h)T-𝐔^h〉∂⁡Ω+λ⁢〈𝐄T-𝐔^h,(𝐰h)T〉∂⁡Ω-𝐢⁢κ⁢(𝐄-𝐔h,∇⁡rh)𝒯h.

By Young’s inequality, we have

∥𝐄T-𝐔^h∥0,∂⁡Ω2≲∥(𝐙h-𝐙~hc)T∥0,∂⁡Ω2+∥(Πhc⁢𝐄-𝐄)T∥0,∂⁡Ω2

(4.35)+∥(𝐔h)T-𝐔^h∥0,∂⁡Ω2+∥(𝐰h)T∥0,∂⁡Ω2+κ⁢∥𝐄-𝐔h∥0,Ω⁢∥∇⁡rh∥0,Ω.

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

(4.36)∥(𝐰h)T∥0,∂⁡Ω2≲1h⁢∥𝐰h∥0,Ω2≲1h⁢(∥𝐰-𝐰h∥0,Ω2+∥𝐰∥0,Ω2)≲κ2⁢h3⁢M2⁢(𝐟~,𝐠~).

It follows from the orthogonality of the decomposition (4.31), the triangle inequality, (4.12), (4.25) and (4.30) that, for κ⁢h≤1,

(4.37)∥∇⁡rh∥0,Ω≤∥𝐙h-𝐙~hc∥0,Ω+∥𝐙h∥0,Ω≲κ12⁢h32⁢M⁢(𝐟~,𝐠~).

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^h∥0,∂⁡Ω≲κ⁢h32⁢M⁢(𝐟~,𝐠~),

The proof is complete. ∎

Now we are ready to state our optimal error estimate for ∥𝐄-𝐔h∥0,Ω.

Theorem 4.6

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

(4.38)∥𝐄-𝐔h∥0,Ω≲κ⁢h2⁢M⁢(𝐟~,𝐠~).

Proof.

If the mesh size h satisfies κ⁢h≥1, then (4.38) follows immediately from (4.25). If κ⁢h≤1, 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.

Theorem 4.7

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

(4.39)∥𝐄-𝐄h∥0,Ω≲(κ⁢h2+Cstab⁢κ2⁢h2)⁢M⁢(𝐟~,𝐠~),

(4.40)κ⁢∥𝐐-𝐐h∥0,Ω≲(κ⁢h+Cstab⁢κ3⁢h2)⁢M⁢(𝐟~,𝐠~).

Moreover, if κ3⁢h2≲1, then

(4.41)∥𝐄h∥0,Ω+∥𝐐h∥0,Ω≲∥𝐟∥0,Ω+∥𝐠∥0,∂⁡Ω,

(4.42)∥𝐄-𝐄h∥0,Ω≲(κ⁢h2+κ2⁢h2)⁢M⁢(𝐟~,𝐠~),

(4.43)κ⁢∥𝐐-𝐐h∥0,Ω≲(κ⁢h+κ3⁢h2)⁢M⁢(𝐟~,𝐠~).

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

(𝐢⁢κ⁢ϵh𝐐,𝐫)𝒯h+(ϵh𝐄,curl⁡𝐫)𝒯h-〈ϵh𝐄^,𝐧×𝐫〉∂⁡𝒯h=0,

-(𝐢⁢κ⁢ϵh𝐄,𝐯)𝒯h+(ϵh𝐐,curl⁡𝐯)𝒯h+〈𝐧×ϵ^h𝐐,𝐯T〉∂⁡𝒯h=-2⁢𝐢⁢κ⁢(𝐄-𝐔h,𝐯)𝒯h,

〈𝐧×ϵ^h𝐐-λ⁢ϵh𝐄^,𝝁〉∂⁡Ω=0,

〈𝐧×ϵ^h𝐐,𝝁〉∂⁡𝒯h\∂⁡Ω=0

for all 𝐫,𝐯∈𝐕h and 𝝁∈𝐌h. 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

∥ϵh𝐄∥0,Ω+∥ϵh𝐐∥0,Ω≲Cstab⁢κ⁢∥𝐄-𝐔h∥0,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=1κ⁢h. The implementation is done using the MATLAB software package iFEM (cf. [5]).

$Figure 1 Test 1. The relative errors ∥𝐐-𝐐h∥0,Ω/∥𝐐∥0,Ω${\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-𝐐h∥0,Ω/∥𝐐∥0,Ω and ∥𝐄-𝐄h∥0,Ω/∥𝐄∥0,Ω of the HDG solution for κ=10,20,30.

Test 1

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=(e𝐢⁢κ⁢z,e𝐢⁢κ⁢x,e𝐢⁢κ⁢y)T.

For fixed wave number κ, we first compute the relative errors ∥𝐐-𝐐h∥0,Ω/∥𝐐∥0,Ω and ∥𝐄-𝐄h∥0,Ω/∥𝐄∥0,Ω, respectively. The left graph in Figure 1 displays the relative errors ∥𝐐-𝐐h∥0,Ω/∥𝐐∥0,Ω of the HDG solution for κ=10,20,30, while the right graph shows the same errors for ∥𝐄-𝐄h∥0,Ω/∥𝐄∥0,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.

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

$Figure 3 Test 1. Left: The traces of the real part of the first component of the HDG solution for κ=16${\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-𝐐h∥0,Ω/∥𝐐∥0,Ω and ∥𝐄-𝐄h∥0,Ω/∥𝐄∥0,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 κ⁢h=2. We observe that the relative errors cannot be controlled by κ⁢h 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 κ3⁢h2=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=0.5,0≤z≤1 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 κ⁢h=1.

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

Test 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⁡(κ⁢y)⁢J0⁢(κ⁢r),cos⁡(κ⁢z)⁢J0⁢(κ⁢r),𝐢⁢κ⁢J0⁢(κ⁢r))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 div⁡𝐟≠0.

The left graph in Figure 4 displays the relative errors ∥𝐐-𝐐h∥0,Ω/∥𝐐∥0,Ω and ∥𝐄-𝐄h∥0,Ω/∥𝐄∥0,Ω when the mesh condition κ⁢h=2 is enforced. It shows that the relative errors can not be controlled by κ⁢h 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 κ3⁢h2=2. It clearly indicates that the relative errors can be controlled under this mesh condition.

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11401417

Award Identifier / Grant number: 11225107

Funding statement: 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).

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Received: 2015-10-23

Revised: 2016-1-11

Accepted: 2016-3-1

Published Online: 2016-6-1

Published in Print: 2016-7-1

Articles in the same Issue

https://doi.org/10.1515/cmam-2016-0021

Keywords for this article

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