hybridizable discontinuous galerkin methods for timoshenko beams

J Sci Comput (2010) 44: 1–37DOI 10.1007/s10915-010-9357-2

Hybridizable Discontinuous Galerkin Methodsfor Timoshenko Beams

Fatih Celiker · Bernardo Cockburn · Ke Shi

Received: 13 October 2009 / Revised: 16 February 2010 / Accepted: 18 February 2010 /Published online: 7 March 2010© Springer Science+Business Media, LLC 2010

Abstract In this paper, we introduce a new class of discontinuous Galerkin methods forTimoshenko beams. The main feature of these methods is that they can be implemented inan efficient way through a hybridization procedure which reduces the globally coupled un-knowns to approximations to the displacement and bending moment at the element bound-aries. After displaying the methods, we obtain conditions under which they are well defined.We then compare these new methods with the already existing discontinuous Galerkin meth-ods for Timoshenko beams. Finally, we display extensive numerical results to ascertain theinfluence of the stabilization parameters on the accuracy of the approximation. In particular,we find specific choices for which all the variables, namely, the displacement, the rotation,the bending moment and the shear force converge with the optimal order of k +1 when eachof their approximations are taken to be piecewise polynomial of degree k ≥ 0.

Keywords Discontinuous Galerkin methods · Timoshenko beams · Hybridization

The research of the second author was partially supported by the National Science Foundation (GrantDMS-0712955) and by the Minnesota Supercomputing Institute.

F. Celiker (�)Department of Mathematics, Wayne State University, Detroit, MI 48202, USAe-mail: [email protected]

B. Cockburn · K. ShiSchool of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

B. Cockburne-mail: [email protected]

K. Shie-mail: [email protected]

mailto:[email protected]



2 J Sci Comput (2010) 44: 1–37

1 Introduction

In this paper, we introduce a new class of discontinuous Galerkin methods for Timoshenkobeams called hybridizable discontinuous Galerkin (HDG) methods. The dimensionless formof the Timoshenko beam model [9], is given by the differential equations, see [4],

VV′ = θ − d2 T

GA, θ ′ = M

EI, M ′ = T , T ′ = q, (1.1)

in � := (0,1). Here, the unknowns are the transverse displacement VV, the rotation of thetransverse cross-section of the beam θ , the bending moment M , and the shear force T . Thematerial and geometrical properties of the beam are characterized by the shear modulus G,the cross-section area A, the Young modulus E, and the moment of inertia I . The remainingdata of the problem are the transverse load, q , and the boundary conditions which we taketo be

VV = VVD, θ = θN on ∂� = {0,1}. (1.2)

The parameter 0 < d < 1 represents the thickness of the beam. The case d � 1 is of par-ticular interest from a numerical point of view since we intend to devise methods whoseconvergence properties remain invariant as d goes to zero. We will also consider the limit-ing case in which d = 0. Note that this corresponds to the so called Euler-Bernoulli beammodel, and is a one-dimensional form of the biharmonic problem. The Timoshenko beammodel on the other hand is a one-dimensional form of the Reissner-Mindlin plate model.

To put our results in perspective, let us briefly review the work done on the subject. In [3],DG methods for the Timoshenko beams were introduced and sufficient conditions ensuringthe existence and uniqueness of their approximate solutions were proved. Preliminary nu-merical experiments were obtained which indicated that, when polynomials of degree k areused, the optimal order of convergence of k + 1 is achieved for all variables. Later, in [2],the fact that all the numerical traces of the h-version of the DG method superconverge withorder 2k + 1 was uncovered and a local post-processing resulting in a uniformly accuratesolution of order 2k + 1 was devised and numerically tested. These results hold uniformlywith respect to the thickness of the beam. In [4], these results were put in firm mathematicalground.

On the other hand, the HDG methods were introduced in [7] in the framework of second-order elliptic problems. The main feature of these methods is that their approximate solu-tions can be expressed in an element-by-element fashion in terms of an approximate tracesatisfying a global weak formulation. Since the associated matrix is symmetric and positivedefinite, these methods can be efficiently implemented. In [5], the single-face HDG method(SFH) for second order elliptic problems was introduced. It was proved that by using poly-nomials of degree k ≥ 0 for both the potential as well as the flux, the order of convergencein L2 of both unknowns is k + 1. Later it was shown [8] that many other DG methods,including a wide class of HDG methods, have these optimal convergence properties as well.

In [6], the SFH method was applied to the biharmonic problem �2u = f with the hopethat all variables would converge optimally. However, for this HDG method, optimal con-vergence orders were obtained only for u and ∇u. The approximations to �u and ∇�u

where proven to converge with order k + 12 and k − 1

2 , respectively, and numerical experi-ments confirming the sharpness of these results were obtained. As a first step for uncoveringHDG methods with optimal orders of convergence for all the variables, not only for the bi-harmonic but also for plates and shells, we introduce here a wide class of HDG methods forthe Timoshenko beam.

J Sci Comput (2010) 44: 1–37 3

The rest of the paper is organized as follows. In Sect. 2 we describe the HDG methods,and in Sect. 3 we state conditions under which they are well defined. In Sect. 4, we statea theorem showing that they can be implemented efficiently through an hybridization pro-cedure. In Sect. 5, we compare them with the already existing DG methods. In Sect. 6, wedisplay extensive numerical results to ascertain the influence of the stabilization parameterson the accuracy of their approximations. We end with some concluding remarks in Sect. 7.

2 The HDG Methods

Let us describe the HDG methods under consideration. We begin by introducing our nota-tion. Set

�h := {Ij = (xj−1, xj ) : 0 = x0 < x1 < · · · < xN−1 < xN = 1}.We associate to this partition of the domain � the set of its nodes,

Eh := {x0, x1, . . . , xN },

and the set of its interior nodes E ◦h := Eh\∂�. Moreover, for each mesh element K ∈ �h,

we denote its length by hK and set

h := maxK∈�h

{hK}.

Next, given a polynomial degree k ≥ 0 and an element K ∈ �h, we define Pk(K) asthe set of polynomials of degree less than or equal to k on K. The space of piecewisepolynomials on � is defined accordingly as

V kh := {v : �h �→ R : v|K ∈ Pk(K) for all K ∈ �h}.

We also set

L20(Eh) := {w ∈ L2(Eh) : w = 0 on ∂�}.

Here, L2(Eh) is simply the set of vectors v = (v1, v2, . . . , vN+1) ∈ RN+1 such that∑N+1

i=1 v2i < ∞, i.e. that vi < ∞ for all i = 1,2, . . . ,N + 1.

We are now ready to define the HDG methods. The HDG methods seek an approxima-tion (Th,Mh, θh, VVh, M̂h, V̂Vh) to the exact solution (T ,M,θ, VV,M|Eh

, VV|Eh), in the finite

dimensional space [V kh ]4 × L2(Eh) × L2(Eh) determined by requiring that

−(VVh, v′1)�h

+ 〈V̂Vh, v1 n〉∂�h= (θh, v1)�h

− d2 (Th/GA,v1)�h, (2.1a)

−(θh, v′2)�h

+ 〈θ̂h, v2 n〉∂�h= (Mh/EI, v2)�h

, (2.1b)

−(Mh, v′3)�h

+ 〈M̂h, v3 n〉∂�h= (Th, v3)�h

, (2.1c)

−(Th, v′4)�h

+ 〈T̂h, v4 n〉∂�h= (q, v4)�h

, (2.1d)

〈θ̂h,mn〉∂�h= 〈θN ,mn〉∂�, (2.1e)

〈T̂h,wn〉∂�h= 0, (2.1f)

4 J Sci Comput (2010) 44: 1–37

hold for all (v1, v2, v3, v4,m,w) ∈ [V kh ]4 ×L2(Eh)×L2

0(Eh). Here, the outward unit normalvectors are n(x∓) := ±1 for x ∈ Eh. The “volume” inner product is defined as

(u, v)�h:=

N∑

j=1

(u, v)Ij where (u, v)Ij :=∫

Ij

u(x)v(x) dx,

and the boundary inner product is defined as 〈u,v n〉∂�h:= ∑N

j=1〈u,v n〉∂Ij where〈u,v〉∂Ij := u(x−

j )v(x−j ) + u(x+

j−1)v(x+j−1), and ϕ(x±) := limε↓0 ϕ(x ± ε) for x ∈ Eh.

Note that the boundary condition (1.2) on θ is imposed by (2.1e). The boundary condition(1.2) on VV is imposed as follows:

V̂Vh = VVD on ∂�. (2.2a)

To complete the definition of the HDG method, we need to express the numerical traces T̂h

and θ̂h in terms of the unknowns:

θ̂h = θh − αθ(Mh − M̂h)n − τ(VVh − V̂Vh)n, (2.2b)

T̂h = Th − τ(Mh − M̂h)n + αT (VVh − V̂Vh)n, (2.2c)

where αθ , αT , and τ are non-negative functions defined on ∂�h. They have to be suitablydefined to guarantee the existence and uniqueness of the approximate solution.

3 Existence and Uniqueness of the HDG Solution

In this section we provide sufficient conditions under which the HDG method introduced inthe previous section defines a unique solution. As is usual for DG methods, the existence anduniqueness of the approximation depends strongly on the definition of the numerical traces(2.2), and hence on the parameters αθ , αT , and τ . We state our existence and uniquenessresults in the following two theorems.

Theorem 3.1 Consider the HDG method defined by the weak formulation (2.1), and theformulas (2.2) for the numerical traces. Suppose that the parameter d is strictly positive,and that the stabilization parameters αθ , αT , and τ are non-negative. Then the method hasa unique solution in the following cases:

(1) αT ,αθ > 0 on ∂�h.(2) αT or τ > 0 on at least one point of ∂Ij for all j , and

αθ > 0 on at least one point of ∂�h

(3) αT or τ > 0 on at least one point of ∂Ij for all j , andτ > 0 on at least one point of ∂�h

(4) αT or τ > 0 on at least one point of ∂Ij for all j , andk ≥ 1.

Next, we state an analogous result for the d = 0 case.

Theorem 3.2 Consider the HDG method defined by the weak formulation (2.1), and theformulas (2.2) for the numerical traces. Suppose that the parameter d is equal to zero,namely, we consider HDG approximations to the Euler-Bernoulli beam model. Suppose

J Sci Comput (2010) 44: 1–37 5

further that the stabilization parameters αθ , αT , and τ are non-negative. Then the methodhas a unique solution in the following cases.

(1) αT ,αθ > 0 on ∂�h.(2) αT or τ > 0 on at least one point of ∂Ij for all j , and

αθ > 0 on ∂�.(3) αT or τ > 0 on at least one point of ∂Ij for all j , and

k ≥ 1, andτ > 0 on at least one point of ∂�h.

(4) αT or τ > 0 on at least one point of ∂Ij for all j , andk = 0, andτ(x+

0 ) > 0, τ(x−1 ) = 0, τ(x+

N−1) = 0, τ(x−N) > 0.

Note that it is possible to take τ = 0 on ∂�h. It is also possible to take αT = αθ =0 on ∂�h. Furthermore, we have existence and uniqueness if αT = αθ = 0 on ∂�h, andτ > 0 at only one node of each element. This corresponds to the single-face hybridizable(SFH) method considered in [6] for biharmonic problems. Note that, for the SFH methodsit is required that τ > 0 on ∂�. For this one dimensional case, we were able to relax thiscondition for k ≥ 1, as can be seen from Case (3) of Theorem 3.2.

To retain the clarity of the presentation, we give a proof of Theorems 3.1 and 3.2 inAppendix A.

4 Characterization of the Approximate Solution

In this section, we show that the only globally coupled unknowns of the HDG method de-fined by the weak formulation (2.1), and the formulas (2.2) for the numerical traces arethe approximations at the nodes to the displacement and bending moment given by the nu-merical traces V̂Vh and M̂h, respectively. We also show that the remaining components ofthe approximate solution can be expressed solely in terms of element-by-element-definedoperators acting on V̂Vh and M̂h. To do this, we follow the framework provided in [7].

4.1 The Local Solvers

We begin by introducing the above-mentioned locally defined operators which we call thelocal solvers associated with the method. The first local solver is defined on the elementK ∈ �h as the mapping ω ∈ L2(∂K) �→ (Tω,Mω,�ω,Wω) ∈ [Pk(K)]4 where

− (Wω,v′1)K + 〈ω,v1 n〉∂K = (�ω,v1)K − d2 (Tω/GA,v1)K , (4.1a)

− (�ω,v′2)K + 〈�̂ω, v2 n〉∂K = (Mω/EI, v2)K , (4.1b)

− (Mω,v′3)K = (Tω,v3)K, (4.1c)

− (Tω,v′4)K + 〈̂Tω,v4 n〉∂K = 0 (4.1d)

for all vi ∈ Pk(K) for i = 1,2,3,4. Here,

�̂ω = �ω − αθMωn − τ(Wω − ω)n, (4.2a)

T̂ω = Tω − τMωn + αT (Wω − ω)n. (4.2b)

6 J Sci Comput (2010) 44: 1–37

The second local solver is defined on the element K ∈ �h as the mapping μ ∈ L2(∂K) �→(Tμ,Mμ,�μ,Wμ) ∈ [Pk(K)]4 where

− (Wμ,v′1)K = (�μ,v1)K − d2 (Tμ/GA,v1)K , (4.3a)

− (�μ,v′2)K + 〈�̂μ, v2 n〉∂K = (Mμ/EI, v2)K , (4.3b)

− (Mμ,v′3)K + 〈μ,v3 n〉∂K = (Tμ,v3)K, (4.3c)

− (Tμ,v′4)K + 〈̂Tμ,v4 n〉∂K = 0 (4.3d)


�̂μ = �μ − αθ(Mμ − μ)n − τWμn, (4.4a)

T̂μ = Tμ − τ(Mμ − μ)n + αT Wμn. (4.4b)

The third local solver is defined on the element K ∈ �h as the mapping q ∈ L2(K) �→(Tq,Mq,�q,Wq) ∈ [Pk(K)]4 where

− (Wq, v′1)K = (�q, v1)K − d2 (Tq/GA,v1)K , (4.5a)

− (�q, v′2)K + 〈�̂q, v2 n〉∂K = (Mq/EI, v2)K , (4.5b)

− (Mq, v′3)K = (Tq, v3)K, (4.5c)

− (Tq, v′4)K + 〈̂Tq, v4 n〉∂K = (q, v4)K (4.5d)


�̂q = �q − αθMq n − τWq n, (4.6a)

T̂q = Tq − τMq n + αT Wq n. (4.6b)

4.2 The Characterization of the HDG Solution for k ≥ 1

The function VVD, as well as any other function defined only on ∂� = {0,1} is extended toEh by zero. We also set

ωh : ={

V̂Vh on ∂�h\∂�

0 on ∂�,(4.7a)

so that we have that V̂Vh = ωh + VVD where ωh ∈ L20(Eh). Also, to simplify the notation we

write

μh : = M̂h on ∂�h. (4.7b)

We can now state a characterization of the approximate solution in terms of the local solvers.

Theorem 4.1 Suppose that the conditions of Theorem 3.1 (Theorem 3.2, if d = 0) aresatisfied. Suppose also that k ≥ 1. Then the approximate solution (Th, Mh, θh, VVh, μh,ωh) ∈ [V k

h ]4 × L2(Eh) × L20(Eh) given by the HDG method can be expressed in terms of the

J Sci Comput (2010) 44: 1–37 7

local solvers as

Th = Tωh + TVVD + Tμh + Tq, (4.8a)

Mh = Mωh + MVVD + Mμh + Mq, (4.8b)

θh = �ωh + �VVD + �μh + �q, (4.8c)

VVh = Wωh + WVVD + Wμh + Wq, (4.8d)

where (μh,ωh) ∈ L2(Eh) × L20(Eh) satisfies

ah(μh,m) + bh(ωh,m) = ��h (m), (4.9a)

bh(w,μh) = �Th (w) (4.9b)

for all m ∈ L2(Eh) and w ∈ L20(Eh). Here

ah(μ,m) = (Mμ/EI,Mm)�h+ d2(Tμ/GA,Tm)�h

+ 〈1, αθ (μ − Mμ)(m − Mm)〉∂�h+ 〈1, αT WμWm〉∂�h

,

bh(ω,m) = (�ω,Tm)�h,

��h (m) = −(q,Wm)�h

− (�VVD,Tm)�h+ 〈θN ,mn〉∂�,

�Th (w) = −(q,Ww)�h

.

Next, let us briefly discuss this result whose detailed proof can be found in Appendix B.Note that the system of equations for the numerical traces μh and ωh is of the form

[Ah Bh

Bth 0

] [[μh][ωh]

]

=[

bh,1

bh,2

]

. (4.10)

This is a reflection of the fact that μh approximates the bending moment M , that ωh approx-imates the displacement VV, and that we can write our original problem (1.1) as

[I

EI− d

dx

(d2

GAddx

) − d2

dx2

− d2

dx2 0

] [M

VV

]

=[

0−q

]

.

Let us illustrate this in the case in which q , GA, and EI are constants, and we take a uniformgrid of meshsize h := 1/N and a polynomial degree k ≥ 3. In this case, we obtain that

Ah = 1

6EI

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

2 11 4 1

1 4. . .

1. . . 1. . . 4 1

1 2

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

− d2

h2 GA

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−1 11 −2 1

1 −2. . .

1. . . 1. . . −2 1

1 −1

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

8 J Sci Comput (2010) 44: 1–37

and

Bh = − 1

h2

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1−2 11 −2 1

1 −2. . .

1. . . 1. . . −2

1

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

where Ah is a square matrix of order N + 1 and Bh an (N + 1)× (N − 1) matrix. Moreover,

b1,h =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−VVD(x0)

VVD(x0)

0...

0VVD(xN)

−VVD(xN)

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+ hq

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

11.........

11

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+ h

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

θD(x0)

00...

00

θD(xN)

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, b2,h = −q

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1/211...

11

1/2

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

where b1,h and b2,h are 1 × (N + 1) and 1 × (N − 1) matrices, respectively.We immediately recognize that the matrix Ah is a finite difference approximation of the

operator IEI

− ddx

( d2

GAddx

) and that the matrix Bh is a finite difference approximation of − d2

dx2 .Similarly, we see that b1,h is an approximation of 0 and b2,h an approximation of −q . Finally,note that all of the above matrices are independent of the stabilization functions τ , αθ and αT

as well as of the polynomial degree k. This is a reflection of the fact that the approximationμh and ωh is in fact exact. In other words, in this case, (4.10) is nothing but the matrixequation determining the values of the exact bending moment and displacement at the nodes.

4.3 Characterization of the HDG Solution for k = 0

To end this section, we would like to note that the above characterization theorem is validonly for k ≥ 1. This is not because the approach to obtain this result is different for k = 0,but only because the local solvers take a form which does not allow for some simplificationsthat take place in the case k ≥ 1. For k = 0, the characterization of the approximation isgiven in the next result.

Theorem 4.2 Suppose that the conditions of Theorem 3.1 (Theorem 3.2, if d = 0) aresatisfied. Suppose also that k = 0. Then the approximate solution (Th, Mh, θh, VVh, μh,ωh) ∈ [V 0

h ]4 × L2(Eh) × L20(Eh) given by the HDG method can be expressed in terms of the

local solvers as

Th = Tωh + TVVD + Tμh + Tq,

Mh = Mωh + MVVD + Mμh + Mq,

θh = �ωh + �VVD + �μh + �q,

VVh = Wωh + WVVD + Wμh + Wq,

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where (μh,ωh) ∈ L2(Eh) × L20(Eh) satisfies

a�h (μh,m) + b�

h (ωh,m) = ��h (m), (4.11a)

aTh (μh,w) + bT

h (ωh,w) = �Th (w) (4.11b)

for all m ∈ L2(Eh) and w ∈ L20(Eh). Here

a�h (μ,m) = (Mμ/EI,Mm)�h

+ d2(Tμ/GA,Tm)�h

+ 〈1, αT WμWm〉∂�h+ 〈1, αθ (μ − Mμ)(m − Mm)〉∂�h

,

b�h (ω,m) = (Mω/EI,Mm)�h

+ 〈ω, (Tm)n〉∂�h

+ 〈1, αT WωWm〉∂�h+ 〈1, (τω − αθMω)(m − Mm)〉∂�h

,

aTh (μ,w) = −(Mμ/EI,Mw)�h

+ 〈μ, (�w)n〉∂�h

− 〈1, αθMμMw〉∂�h+ 〈1, (τμ + αT Wμ)(w − Ww)〉∂�h

,

bTh (ω,w) = −(Mω/EI,Mw)�h

− 〈1, αθMωMw〉∂�h

− 〈1, αT (ω − Wω)(w − Ww)〉∂�h,

and

��h (m) = −(q,Wm)�h

− 〈VVD, (̂Tm)n〉∂� + 〈θN ,mn〉∂�,

�Th (w) = −(q,Ww)�h

− 〈VVD, (̂Tw)n〉∂�.

The proof of this result easily follows from Lemma B.2 in Appendix B.

5 Relationship Between the HDG Methods and Other DG Methods

In this section, we investigate the relationship between the HDG methods introduced in thispaper with the DG methods for Timoshenko beams introduced in [3], and analyzed in [4]and [1].

These methods were defined by the weak formulation, (2.1a)–(2.1d). The conservativityconditions (2.1e) and (2.1f) were not taken into account explicitly, because they were au-tomatically satisfied by the definition of the numerical traces which we give next. For aninterior node x ∈ E ◦

h they are defined by

V̂Vh = {{ VVh }} + C11 �VVh� + C12 �θh� + C13 �Mh� + C14 �Th�, (5.1a)

θ̂h = {{θh}} + C21 �VVh� + C22 �θh� + C23 �Mh� + C24 �Th�, (5.1b)

M̂h = {{Mh}} + C31 �VVh� + C32 �θh� + C33 �Mh� + C34 �Th�, (5.1c)

T̂h = {{Th }} + C41 �VVh� + C42 �θh� + C43 �Mh� + C44 �Th�, (5.1d)

where the average, {{·}}, and the jump, �·�, operators are defined by

{{ϕ}}(x) := 1

2(ϕ(x+) + ϕ(x−)), �ϕ�(x) := ϕ(x−) − ϕ(x+).

10 J Sci Comput (2010) 44: 1–37

At x = 0, the numerical traces of the DG methods are defined by

V̂Vh(0) = VVD(0), (5.2a)

θ̂h(0) = θN(0), (5.2b)

M̂h(0) = Mh(0+) + C31(0)(VVD(0) − VVh(0

+)) + C32(0)(θN(0) − θh(0+)), (5.2c)

T̂h(0) = Th(0+) + C41(0)(VVD(0) − VVh(0

+)) + C42(0)(θN(0) − θh(0+)). (5.2d)

And at x = 1,

V̂Vh(1) = VVD(1), (5.3a)

θ̂h(1) = θN(1), (5.3b)

M̂h(1) = Mh(1−) + C31(1)(VVh(1

−) − VVD(1)) + C32(1)(θh(1−) − θN(1)), (5.3c)

T̂h(1) = Th(1−) + C41(1)(VVh(1

−) − VVD(1)) + C42(1)(θh(1−) − θN(1)). (5.3d)

The following theorem provides a recipe for choosing the coefficients Cij , 1 ≤ i, j ≤ 4,so that the solution provided by an HDG method is identical to the one provided by thepreviously defined DG methods. We emphasize the fact we are interested in such an identi-fication only for theoretical purposes because the HDG methods are computationally muchmore efficient than the DG methods as we have shown in Sect. 4.

Let us introduce some notation that will be used in the next theorem. For any two func-tions ϕ and α which are possibly double valued on E ◦

h we define

�ϕ�α = ϕ−α+ − ϕ+α−.

Let us also note that

�ϕn� = (ϕn)− − (ϕn)+ = ϕ−n− − ϕ+n+ = ϕ− + ϕ+,

and, similarly, that

�ϕn�α = ϕ−α+ + ϕ+α−.

We are now ready to state our result.

Theorem 5.1 Let (T Hh ,MH

h , θHh , VV

Hh , M̂h, V̂Vh) be the solution of the HDG method defined

by the weak formulation (2.1) where the numerical traces V̂Vh, θ̂h and T̂h satisfy (2.2). Sup-pose that (Th,Mh, θh, VVh) is the solution of the DG method defined by the weak formulation(2.1a)–(2.1d), and the formulas (5.1)–(5.3) for the numerical traces. Then

(T Hh ,MH

h , θHh , VV

Hh ) = (Th,Mh, θh, VVh)

provided that

C11 = (�τn��τ � + �αθn��αT �)/2D,

C12 = −�τn�/D,(5.4)

C13 = −�τ �αθ/D,

C14 = �αθn�/D,

J Sci Comput (2010) 44: 1–37 11

C21 = −(�αθαT �τ + �τ 2n�τ )/D,

C22 = −(�τn��τ � + �αT n��αθ �)/2D,

(5.5)C23 = −(�αθn�τ2 + �αθαT n�αθ

)/D,

C24 = −C13,

C31 = �τ �αT/D,

C32 = −�αT n�/D

(5.6)C33 = −C22,

C34 = C12,

C41 = (�αT n�τ2 + �αT αθn�αT)/D,

C42 = −C31,(5.7)

C43 = C21,

C44 = −C11,

at the interior nodes x ∈ E ◦h , and

C31 = τn/αθ ,

C32 = −1/αθ ,

C41 = (τ 2 + αT αθ )/αθ ,

C42 = −C31,

(5.8)

at ∂�. Here

D := �τn�2 + �αT n��αθn�,

and we assume that D > 0 on E ◦h , and αθ > 0 on ∂�.

A detailed proof of this theorem can be found in Appendix C. Next, we discuss threeapplications of this result.

• An optimally convergent DG method was studied by Celiker et al. in [4]. This methodis obtained by setting

C11 = C22 = −C33 = −C44 = 1/2,

at all interior nodes,

−C32(1) = C41(1) = 1

hN

,

and all of the remaining coefficients to zero. It is not difficult to see that it is impossibleto express this DG method as an HDG method. In other words, it is impossible to pick theparameters τ , αT , and αθ in (2.2) so that the resulting numerical traces coincide with thenumerical traces of the DG method above. Let us see why this is so. For this DG method

12 J Sci Comput (2010) 44: 1–37

C12(x) = 0 at all interior nodes x ∈ E ◦h . By Theorem 5.1 C12 = �τn�/D = (τ− + τ+)/D.

This implies that τ− + τ+ = 0, and hence that τ− = τ+ = 0. Then

D = �τn�2 + �αT n��αθn� = �αT n��αθn� = (α−T + α+

T )(α−θ + α+

θ ),

and since we have assumed D > 0 at the interior nodes, we must have that

α−T or α+

T > 0 and α−θ or α+

θ > 0, (5.9)

at all interior nodes. On the other hand, C14 = 0 implies by Theorem 5.1 that �αθn�/D = 0,and hence α−

θ + α+θ = 0. But this is possible only if α−

θ = α+θ = 0, which contradicts the

second part of (5.9).

• Another family of DG method studied in Celiker [1] is obtained as follows. Let c be anarbitrary positive number independent of the mesh-size h, and suppose that

C14(x) = −C23(x) = −C32(x) = C41(x) = c ∀x ∈ Eh. (5.10)

Suppose further that

(Cii(x) − 1/2)2 ≤ c for i = 1,2,3,4,

C212(x), C2

13(x), C221(x), C2

24(x), C231(x), C2

34(x), C242(x), C2

43 ≤ c, (5.11)

for all x ∈ Eh. It was shown in [1] that these methods converge optimally in L2(�h)-normto the exact solution for all the unknowns. By a straightforward computation we can showthat these DG methods can be expressed as HDG methods if we take

c = 1

2,

and

τ = 0, αT = αθ = 1

at all interior nodes, and

τ = 0, αT = 1

2, αθ = 2

at the boundary nodes. It is easy to see that with this choice of the parameters the conditionsof Theorem 5.1 are satisfied, and hence the resulting DG method is identical to the HDGmethod.

• The one-dimensional version of the SFH method for the biharmonic [6] takes αT =αθ = 0 on ∂�h, and τ > 0 at one end and zero at the other end of every subinterval. It is alsorequired that for the subintervals which intersect ∂� we should take τ > 0. We can thus take

τ(x+0 ) > 0, τ (x+

j ) = 0 for j = 1,2, . . . ,N − 1,

τ (x−1 ) = 0, τ (x−

j ) > 0 for j = 2,3, . . . ,N.(5.12)

In particular, τ(x−1 ) = τ(x+

1 ) = 0 and as a consequence, D(x1) = �τn�2(x1) = 0. Therefore,by Theorem 5.1, we see that it is impossible to identify this SFH method with a previouslyintroduced DG method. A similar argument can be used to show that this is true for anyother SFH method.

J Sci Comput (2010) 44: 1–37 13

6 Numerical Results

In this section, we display numerical results showing the performance of the numerical meth-ods considered in this paper. We solve (1.1) with

q(x) = ex, (EI)(x) = ex, (GA)(x) = e−x,

together with the boundary conditions

VV(0) = VV(1) = θ(0) = θ(1) = 0.

To study the effect of the penalization parameters τ , αT , and αθ on the performance of themethod, we display results for a variety of choices of these parameters. Also, to find out ifthe HDG method is free from shear locking, the thickness of the beam, d , is taken be 10−2

and then decreased down to 10−8.We display our numerical results in Tables 1 through 10. In Table 1, we display re-

sults corresponding to the HDG method obtained by setting αT = αθ = τ = 1 on ∂�h.Therein, the column labeled k denotes the polynomial degree used for the approximation,and “mesh = i” means we employed a uniform mesh with 2i elements. We also displaynumerical orders of convergence which are computed as follows. Let eu(i) denote the errorwhere a mesh with 2i elements has been employed to obtain the HDG solution. Then theorder of convergence, ri , at level i is defined as

ri := log( eu(i−1)

eu(i))

log 2.

The first part of the table shows the numerical results for the problem (1.1) with d = 10−2,and the second part for d = 10−8. We see that the method converges optimally for all thevariables. We also notice that the convergence is independent of the parameter d . This indi-cates that the method is free from shear locking.

In Table 2, we carry out a similar history of convergence study for the SFH method(5.12). We see that the method converges optimally for all the variables, and the convergenceremains invariant as we decrease d . It is worth noting that we do not see the suboptimal con-vergence of the same method that was proved, and observed numerically, for the biharmonicproblems in two-space dimensions [6].

In order to be able to display an extensive set of numerical results, in Tables 3 through 10we report only a summary of what we have observed as a result of our study of the historyof convergence. In these tables the column k again shows the polynomial degree of the ap-proximation, and the columns ‖eu‖L2(�), for u = T ,M,θ, VV shows the order of convergencewe have observed as a result of the numerical experiments whose details we suppress. Thedash “−” means that the method failed to converge for that particular combination of thepolynomial degree and the choice of the stabilization parameters.

From Table 3 we see that whenever αT ≡ 0, αθ ≡ 0, and τ identically equal to a constanton ∂�h, the only choice which converges optimally for all the variables is obtained by settingτ ≡ O(1). Table 5 shows that in an analogous situation we have a little more flexibility ifwe employ an SFH method. That is, if we set αT ≡ 0, αθ ≡ 0, and τ− ≡ 0 then we observeoptimal convergence for τ+ = 1/hμ if μ = 0,1,2.

In Table 4 we consider the other extreme, namely, we take τ ≡ 0, and αT = αθ ≡ ν

on ∂�h for some constant ν ≥ 0. Once again, we see that the only choice to get optimalconvergence for all the unknowns is obtained by setting ν = O(1).

14 J Sci Comput (2010) 44: 1–37

Table 1 τ ≡ 1, αθ ≡ 1, αT ≡ 1

k Mesh ‖eT ‖L2(�)

‖eM‖L2(�)

‖eθ‖L2(�)

‖eVV‖L2(�)

Error Order Error Order Error Order Error Order

d = 10−2

0 5 1.37E-02 0.82 7.34E-03 0.86 7.41E-03 0.94 1.61E-02 0.99

6 7.45E-03 0.88 3.89E-03 0.91 3.78E-03 0.97 8.05E-03 1.00

7 3.90E-03 0.93 2.01E-03 0.95 1.91E-03 0.99 4.03E-03 1.00

8 2.00E-03 0.96 1.03E-03 0.97 9.60E-04 0.99 2.02E-03 1.00

1 3 1.32E-03 1.98 6.63E-04 1.91 1.27E-03 1.92 6.69E-04 2.01

4 3.33E-04 1.99 1.70E-04 1.96 3.27E-04 1.96 1.67E-04 2.00

5 8.36E-05 1.99 4.31E-05 1.98 8.29E-05 1.98 4.16E-05 2.00

6 2.09E-05 2.00 1.08E-05 1.99 2.09E-05 1.99 1.04E-05 2.00

2 3 1.27E-05 2.98 9.93E-06 2.96 1.33E-05 2.98 4.57E-06 2.99

4 1.60E-06 2.99 1.26E-06 2.98 1.68E-06 2.99 5.73E-07 3.00

5 2.00E-07 2.99 1.58E-07 2.99 2.10E-07 3.00 7.16E-08 3.00

6 2.51E-08 3.00 1.98E-08 3.00 2.63E-08 3.00 8.94E-09 3.00

3 3 6.79E-08 3.95 3.87E-08 3.96 1.49E-07 3.96 1.04E-07 3.97

4 4.29E-09 3.98 2.44E-09 3.99 9.46E-09 3.98 6.55E-09 3.99

5 2.69E-10 3.99 1.53E-10 4.00 5.95E-10 3.99 4.11E-10 3.99

6 1.69E-11 4.00 9.55E-12 4.00 3.73E-11 4.00 2.57E-11 4.00

d = 10−8

0 5 1.39E-02 0.81 7.40E-03 0.86 7.42E-03 0.94 1.61E-02 0.99

6 7.55E-03 0.88 3.93E-03 0.91 3.78E-03 0.97 8.05E-03 1.00

7 3.96E-03 0.93 2.03E-03 0.95 1.91E-03 0.99 4.03E-03 1.00

8 2.03E-03 0.96 1.04E-03 0.97 9.61E-04 0.99 2.02E-03 1.00

1 3 1.32E-03 1.98 6.64E-04 1.91 1.28E-03 1.92 6.69E-04 2.01

4 3.33E-04 1.99 1.70E-04 1.96 3.28E-04 1.96 1.67E-04 2.00

5 8.35E-05 1.99 4.31E-05 1.98 8.30E-05 1.98 4.16E-05 2.00

6 2.09E-05 2.00 1.08E-05 1.99 2.09E-05 1.99 1.04E-05 2.00

2 3 1.27E-05 2.98 9.92E-06 2.96 1.34E-05 2.98 4.57E-06 2.99

4 1.60E-06 2.99 1.26E-06 2.98 1.68E-06 2.99 5.73E-07 3.00

5 2.00E-07 2.99 1.58E-07 2.99 2.11E-07 3.00 7.16E-08 3.00

6 2.51E-08 3.00 1.98E-08 3.00 2.64E-08 3.00 8.95E-09 3.00

3 3 6.78E-08 3.95 3.87E-08 3.96 1.49E-07 3.96 1.04E-07 3.97

4 4.29E-09 3.98 2.43E-09 3.99 9.46E-09 3.98 6.55E-09 3.99

5 2.69E-10 3.99 1.53E-10 4.00 5.95E-10 3.99 4.11E-10 3.99

6 1.68E-11 4.00 9.54E-12 4.00 3.73E-11 4.00 2.57E-11 4.00

In Table 6 we carry out a more extensive study of what we have done for Table 1. Wetake τ ≡ αθ ≡ αT ≡ ν for ν = h2, h,1, h−1, h−2. We see that the only choice among theseoptions to get optimal convergence for all the unknowns is the one we considered in Table 1,

J Sci Comput (2010) 44: 1–37 15

Table 2 τ+ ≡ 1, τ− ≡ 0, αθ ≡ 0, αT ≡ 0

k Mesh ‖eT ‖L2(�)

‖eM‖L2(�)

‖eθ‖L2(�)

‖eVV‖L2(�)

Error Order Error Order Error Order Error Order

d = 10−2

0 5 7.09E-02 0.86 4.95E-03 1.27 4.95E-04 0.96 1.17E-03 1.09

6 3.72E-02 0.93 1.98E-03 1.33 2.70E-04 0.88 5.75E-04 1.02

7 1.91E-02 0.97 7.99E-04 1.31 1.43E-04 0.92 2.87E-04 1.00

8 9.64E-03 0.98 3.37E-04 1.24 7.36E-05 0.95 1.44E-04 1.00

1 3 1.94E-03 2.55 3.52E-03 1.78 2.33E-04 2.04 4.66E-04 2.10

4 3.84E-04 2.34 9.43E-04 1.90 5.74E-05 2.02 1.14E-04 2.03

5 8.75E-05 2.13 2.44E-04 1.95 1.43E-05 2.00 2.83E-05 2.01

6 2.13E-05 2.04 6.20E-05 1.98 3.58E-06 2.00 7.07E-06 2.00

2 3 1.31E-05 3.00 3.42E-05 2.81 8.41E-06 2.91 1.59E-05 2.92

4 1.64E-06 3.00 4.57E-06 2.90 1.08E-06 2.96 2.03E-06 2.97

5 2.04E-07 3.00 5.90E-07 2.95 1.37E-07 2.98 2.57E-07 2.98

6 2.55E-08 3.00 7.50E-08 2.98 1.73E-08 2.99 3.23E-08 2.99

3 3 9.87E-08 3.99 2.58E-07 3.81 1.24E-07 3.98 1.89E-07 4.04

4 6.16E-09 4.00 1.72E-08 3.90 7.81E-09 3.99 1.17E-08 4.02

5 3.84E-10 4.00 1.11E-09 3.95 4.89E-10 4.00 7.24E-10 4.01

6 2.40E-11 4.00 7.07E-11 3.98 3.06E-11 4.00 4.51E-11 4.00

d = 10−8

0 5 7.11E-02 0.86 4.95E-03 1.27 4.98E-04 0.95 1.16E-03 1.09

6 3.73E-02 0.93 1.97E-03 1.33 2.72E-04 0.87 5.74E-04 1.02

7 1.91E-02 0.97 7.96E-04 1.31 1.44E-04 0.92 2.87E-04 1.00

8 9.67E-03 0.98 3.36E-04 1.25 7.43E-05 0.95 1.43E-04 1.00

1 3 1.94E-03 2.55 3.52E-03 1.78 2.33E-04 2.04 4.66E-04 2.09

4 3.85E-04 2.34 9.43E-04 1.90 5.73E-05 2.02 1.14E-04 2.03

5 8.75E-05 2.14 2.44E-04 1.95 1.43E-05 2.00 2.83E-05 2.01

6 2.13E-05 2.04 6.20E-05 1.98 3.57E-06 2.00 7.06E-06 2.00

2 3 1.31E-05 3.00 3.42E-05 2.81 8.40E-06 2.91 1.59E-05 2.92

4 1.64E-06 3.00 4.57E-06 2.90 1.08E-06 2.96 2.03E-06 2.97

5 2.04E-07 3.00 5.90E-07 2.95 1.37E-07 2.98 2.57E-07 2.98

6 2.55E-08 3.00 7.50E-08 2.98 1.73E-08 2.99 3.23E-08 2.99

3 3 9.87E-08 3.99 2.58E-07 3.81 1.24E-07 3.98 1.89E-07 4.04

4 6.16E-09 4.00 1.72E-08 3.90 7.80E-09 3.99 1.16E-08 4.02

5 3.84E-10 4.00 1.11E-09 3.95 4.89E-10 4.00 7.23E-10 4.01

6 2.40E-11 4.00 7.07E-11 3.98 3.06E-11 4.00 4.51E-11 4.00

namely, take ν = 1. All the other options result in suboptimal convergence for either M andVV, or for T and θ .

In all of the Tables 1 through 6 we have considered cases where αT ≡ αθ . Therefore,in Tables 7 to 10 we consider possible cases in which these two parameters take different

16 J Sci Comput (2010) 44: 1–37

Table 3 αθ ≡ 0, αT ≡ 0k ‖eT ‖

L2(�)‖eM‖

L2(�)‖eθ ‖

L2(�)‖eVV‖

L2(�)

τ ≡ h2

0 – – – –1 k – k –2 k + 1 k − 1 k –3 k + 1 k − 1 k k − 2

τ ≡ h

0 – – – –1 k + 1 k k + 1 k

2 k + 1 k k + 1 k

3 k + 1 k k + 1 k

τ ≡ 10 k + 1 k + 1 k + 1 k + 11 k + 1 k + 1 k + 1 k + 12 k + 1 k + 1 k + 1 k + 13 k + 1 k + 1 k + 1 k + 1

τ ≡ 1/h

0 – – – –1 k k + 1 k k + 12 k k + 1 k k + 13 k k + 1 k k + 1

τ ≡ 1/h2

0 – – – –1 k k + 1 k k + 12 k k + 1 k k + 13 k k + 1 k k + 1

Table 4 τ ≡ 0k ‖eT ‖

L2(�)‖eM‖

L2(�)‖eθ ‖

L2(�)‖eVV‖

L2(�)

αθ ≡ αT ≡ h2

0 – – – –1 k + 1 k k + 1 –2 k + 1 k k + 1 k − 13 k + 1 k k + 1 k − 1

αθ ≡ αT ≡ h

0 – – – –1 k + 1 k k + 1 k

2 k + 1 k k + 1 k

3 k + 1 k k + 1 k

αθ ≡ αT ≡ 10 k + 1 k + 1 k + 1 k + 11 k + 1 k + 1 k + 1 k + 12 k + 1 k + 1 k + 1 k + 13 k + 1 k + 1 k + 1 k + 1

αθ ≡ αT ≡ 1/h

0 – – – –1 k k + 1 k k + 12 k k + 1 k k + 13 k k + 1 k k + 1

αθ ≡ αT ≡ 1/h2

0 – – – –1 k k + 1 k k + 12 k k + 1 k k + 13 k k + 1 k k + 1

J Sci Comput (2010) 44: 1–37 17

Table 5 τ− ≡ 0, αθ ≡ 0,αT ≡ 0

k ‖eT ‖L2(�)

‖eM‖L2(�)

‖eθ ‖L2(�)

‖eVV‖L2(�)

τ+ ≡ h2

0 – – – –1 k – k –2 k + 1 k − 1 k –3 k + 1 k − 1 k k − 2

τ+ ≡ h

0 – – – –1 k + 1 k k + 1 k

2 k + 1 k k + 1 k

3 k + 1 k k + 1 k

τ+ ≡ 10 k + 1 k + 1 k + 1 k + 11 k + 1 k + 1 k + 1 k + 12 k + 1 k + 1 k + 1 k + 13 k + 1 k + 1 k + 1 k + 1

τ+ ≡ 1/h

0 k + 1 k + 1 k + 1 k + 11 k + 1 k + 1 k + 1 k + 12 k + 1 k + 1 k + 1 k + 13 k + 1 k + 1 k + 1 k + 1

τ+ ≡ 1/h2


Table 6 τ ≡ αθ ≡ αT ≡ νk ‖eT ‖

L2(�)‖eM‖

L2(�)‖eθ ‖

L2(�)‖eVV‖

L2(�)

ν = h2

0 – – – –1 k + 1 k k + 1 –2 k + 1 k k + 1 k − 13 k + 1 k k + 1 k − 1

ν = h

0 – – – –1 k + 1 k k + 1 k

2 k + 1 k k + 1 k

3 k + 1 k k + 1 k

ν = 10 k + 1 k + 1 k + 1 k + 11 k + 1 k + 1 k + 1 k + 12 k + 1 k + 1 k + 1 k + 13 k + 1 k + 1 k + 1 k + 1

ν = 1/h

0 – – – –1 k k + 1 k k + 12 k k + 1 k k + 13 k k + 1 k k + 1

ν = 1/h2

0 – – – –1 k k + 1 k k + 12 k k + 1 k k + 13 k k + 1 k k + 1

18 J Sci Comput (2010) 44: 1–37


L2(�)‖eM‖

L2(�)‖eθ ‖

L2(�)‖eVV‖

L2(�)

τ = h2

0 – – – –1 k – k –2 k + 1 k − 1 k –3 k + 1 k − 1 k –

τ = h

0 – – – –1 k + 1 k k + 1 –2 k + 1 k k + 1 k − 13 k + 1 k k + 1 k − 1

τ = 10 k + 1 k + 1 k + 1 k + 11 k + 1 k + 1 k + 1 k + 12 k + 1 k + 1 k + 1 k + 13 k + 1 k + 1 k + 1 k + 1

τ = 1/h

0 – – – –1 k k + 1 k k + 12 k k + 1 k k + 13 k k + 1 k k + 1

τ = 1/h2

0 – – – –1 k k + 1 k k + 12 k k + 1 k k + 13 k k + 1 k k + 1


L2(�)‖eM‖

L2(�)‖eθ ‖

L2(�)‖eVV‖

L2(�)

τ = h2

0 – – – –1 k + 1 k k + 1 k + 12 k + 1 k k + 1 k + 13 k + 1 k k + 1 k + 1

τ = h

0 – – – –1 k + 1 k k + 1 k + 12 k + 1 k k + 1 k + 13 k + 1 k k + 1 k + 1

τ = 10 k + 1 k + 1 k + 1 k + 11 k + 1 k + 1 k + 1 k + 12 k + 1 k + 1 k + 1 k + 13 k + 1 k + 1 k + 1 k + 1

τ = 1/h

0 – – – –1 k k + 1 k k + 12 k k + 1 k k + 13 k k + 1 k k + 1

τ = 1/h2

0 – – – –1 k k + 1 k k + 12 k k + 1 k k + 13 k k + 1 k k + 1

J Sci Comput (2010) 44: 1–37 19

Table 9 αT ≡ 0, τ ≡ 1k ‖eT ‖

L2(�)‖eM‖

L2(�)‖eθ ‖

L2(�)‖eVV‖

L2(�)

αθ = h2


αθ = h


αθ = 10 k + 1 k + 1 k + 1 k + 11 k + 1 k + 1 k + 1 k + 12 k + 1 k + 1 k + 1 k + 13 k + 1 k + 1 k + 1 k + 1

αθ = 1/h

0 – – – –1 k + 1 k + 1 k k

2 k + 1 k + 1 k k

3 k + 1 k + 1 k k

αθ = 1/h2

0 – – – –1 k + 1 k + 1 – –2 k + 1 k + 1 k − 1 k − 13 k + 1 k + 1 k − 1 k − 1

Table 10 αθ ≡ 0, τ ≡ 1k ‖eT ‖

L2(�)‖eM‖

L2(�)‖eθ ‖

L2(�)‖eVV‖

L2(�)

αT = h2


αT = h


αT = 10 k + 1 k + 1 k + 1 k + 11 k + 1 k + 1 k + 1 k + 12 k + 1 k + 1 k + 1 k + 13 k + 1 k + 1 k + 1 k + 1

αT = 1/h

0 – – – –1 k k k + 1 k + 12 k k k + 1 k + 13 k k k + 1 k + 1

αT = 1/h2

0 – – – –1 – k k k

2 k − 1 k k + 1 k + 13 k − 1 k k + 1 k + 1

20 J Sci Comput (2010) 44: 1–37

values. We observe similar convergence behavior to what we already observed in the caseswe considered in Tables 1 through 6. An interesting observation that might be worth notingthat we did not observe any convergence at all for VVh if αθ ≡ 1, αT ≡ 0, and τ = h2, seeTable 7. Another observation we would like to emphasize is that the choice αT ≡ 0, τ ≡ 1converges optimally for αθ = hμ with μ = 0,1,2. We have a similar behavior if we keep τ

the same but switch the roles of αT and αθ . These conclusions are based on Tables 9 and 10.

7 Concluding Remarks

We have introduced a new class of discontinuous Galerkin methods for Timoshenko beamscalled HDG methods and shown that they can be efficiently implemented through an hy-bridization procedure. Our numerical results indicate that it is possible to devise optimallyconvergent HDG methods which are free from shear locking for a variety of choices of thestabilization parameters. A rigorous error analysis of these HDG methods constitutes thesubject of a forthcoming paper.

Appendix A: Proof of the Existence and Uniqueness Results

In this appendix we give proofs of the existence and uniqueness theorems stated in Sect. 3.The proofs are based on the following technical lemmas.

Lemma A.1 Let (Th,Mh, θh, VVh, M̂h, V̂Vh) be the HDG solution defined by the weak formu-lation (2.1), and the formulas (2.2) for the numerical traces. Then we have the followingidentities:

(Mh/EI,Mh)�h+ d2 (Th/GA,Th)�h

− 〈θ̂h − θh, (Mh − M̂h)n〉∂�h+ 〈T̂h − Th, (VVh − V̂Vh)n〉∂�h

= (q, VVh)�h+ 〈θ̂hn, M̂h〉∂� − 〈T̂hn, V̂Vh〉∂�, (A.1)

(θh, θh)�h− 〈θ̂h − θh, (VVh − V̂Vh)n〉∂�h

= d2 (Th/GA,θh)�h− (Mh/EI, VVh)�h

+ 〈θ̂hn, V̂Vh〉∂�, (A.2)

(Th, Th)�h− 〈T̂h − Th, (Mh − M̂h)n〉∂�h

= −(q,Mh)�h+ 〈T̂hn, M̂h〉∂�. (A.3)

Lemma A.2 With the same notation as in Lemma A.1 the following identities hold true

−〈θ̂h − θh, (Mh − M̂h)n〉∂�h+ 〈T̂h − Th, (VVh − V̂Vh)n〉∂�h

= 〈αθ , (Mh − M̂h)2〉∂�h

+ 〈αT , (VVh − V̂Vh)2〉∂�h

, (A.4)

−〈θ̂h − θh, (VVh − V̂Vh)n〉∂�h

= 〈τ, (VVh − V̂Vh)2〉∂�h

+ 〈αθ , (Mh − M̂h)(VVh − V̂Vh)〉∂�h, (A.5)

−〈T̂h − Th, (Mh − M̂h)n〉∂�h

= 〈τ, (Mh − M̂h)2〉∂�h

− 〈αT , (Mh − M̂h)(VVh − V̂Vh)〉∂�h. (A.6)

J Sci Comput (2010) 44: 1–37 21

The proofs of Lemmas A.1 and A.2 are delayed until the end of this section.Before starting to prove Theorems 3.1 and 3.2 we state and prove an auxiliary lemma in

which we collect some intermediate results.

Lemma A.3 Consider the HDG method defined by the weak formulation (2.1), and theformulas (2.2) for the numerical traces. Suppose that the data of the problem is given by

q = 0 in �, VVD = θN = 0 on ∂� (A.7)

and that the stabilization parameters αθ , αT , and τ are non-negative. Then we have

d Th = 0 in �h, (A.8a)

Mh = 0 in �h, (A.8b)

θh = 0 in �h, (A.8c)

αθM̂h = 0 on ∂�h, (A.8d)

αT VVh = 0 on ∂�h, (A.8e)

τVVh = 0 on ∂�h, (A.8f)

V̂Vh = 0 on Eh. (A.8g)

Furthermore, let L�(·) be the Legendre polynomial of order �, and set

ϕ�j (x) = L�

(2x − xj − xj−1

hj

)

for j = 1,2, . . . ,N.

Then, there exist constants aj such that

VVh(x) = ajϕkj (x) on each element Ij ∈ �h. (A.9)

Proof Inserting (A.4) into (A.1), and using (A.7), we obtain

(Mh/EI,Mh)�h+ d2 (Th/GA,Th)�h

+ 〈αθ , (Mh − M̂h)2〉∂�h

+ 〈αT , (VVh − V̂Vh)2〉∂�h

= 0.

Since, αT ,αθ ≥ 0 on ∂�h, we immediately obtain (A.8a), (A.8b), (A.8d), and

αT (VVh − V̂Vh) = 0 on ∂�h. (A.10)

Then, inserting (A.5) into (A.2) we get,

(θh, θh)�h+ 〈τ, (VVh − V̂Vh)

2〉∂�h= 0.

Since, τ ≥ 0 on ∂�h this implies (A.8c), and

τ(VVh − V̂Vh) = 0 on ∂�h. (A.11)

22 J Sci Comput (2010) 44: 1–37

By (A.8a) and (A.8c), (2.1a) reduces to,

−(VVh, v′)�h

+ 〈V̂Vh, vn〉∂�h= 0 for all v ∈ V k

h . (A.12)

Thus, taking v = χIj , the characteristic function of Ij , and varying j = 1,2, . . . ,N we seethat V̂Vh is identically constant on Eh. Since V̂Vh|∂� = VVD = 0, this constant is zero, whichproves (A.8g). Hence, (A.10) implies (A.8e), and (A.11) implies (A.8f).

It remains to prove (A.9). Clearly, (A.8g) further reduces (A.12) to

−(VVh, v′)�h

= 0 for all v ∈ V kh .

Restricting this equation to a single but arbitrary element Ij by taking a test function v withsupport on that element we get

−(VVh, v′)Ij = 0 for all v ∈ P k(Ij ). (A.13)

Let

VVh(x) =k∑

�=0

a�jϕ

�j (x) for some constants a�

j .

If k = 0 then (A.9) is trivially true, so we assume that k ≥ 1. Taking v in (A.13) such thatv′ = ϕ�

j , we see that a�j = 0 for � = 0,1, . . . , k − 1, and (A.9) follows. �

We are now ready to prove Theorems 3.1 and 3.2.

Proof of Theorem 3.1 Due to the linearity of the problem, it is enough show that the onlysolution to (2.1) with data given by (A.7) is

Th = Mh = θh = VVh = 0 in �h, ωh = 0 on E ◦h , μh = 0 on Eh.

In Lemma A.3 we have already proved that Mh = θh = 0 in �h. Since d > 0, (A.8a) impliesTh = 0 in �h. Also since ωh = V̂Vh on E ◦

h , (A.8g) implies that ωh = 0 on E ◦h . Thus it remains

to prove that

VVh = 0, in �h, (A.14a)

μh = 0, on Eh. (A.14b)

We proceed case by case.Case (1): Suppose that αT > 0 on ∂�h. Then by (A.8e) we see that VVh = 0 on ∂�h.

Since ϕkj |∂Ij �= 0, (A.9) implies that aj = 0 for all j = 1,2, . . . ,N . Hence VVh = 0 on �h.

Similarly, suppose that αθ > 0 on ∂�h. Then (A.8d) implies M̂h = 0 on Eh. Since M̂h = μh

by (4.7b) we reach at (A.14b).Case (2): Suppose that αT or τ > 0 on at least one point of ∂Ij for all j . Then by (A.8e)

and (A.8f) VVh = 0 on ∂�h. Proceeding as in Case (1) we get (A.14a). Since Th = Mh = 0in �h, (2.1c) reduces to

〈M̂h, vn〉∂�h= 0 for all v ∈ V k

h .

Thus,

M̂h ={

0 if k ≥ 1,

constant if k = 0.

For the case k = 0, if αθ > 0 for at least one point of ∂�h, then (A.8d) implies (A.14b).

J Sci Comput (2010) 44: 1–37 23

Case (3): The proof of (A.14a) was given in the proof of Case (2). To prove (A.14b)suppose that τ > 0 on at least one point of ∂�h, say τ(x−

i ) > 0. Note that Th = 0 on �h

implies that T̂h is constant on Eh by (2.1d). In particular, T̂h(x+i−1) = T̂h(x

−i ). Recall that by

(A.8b), (A.8g), and (A.14a), the definition of T̂h reduces to

T̂h = τM̂hn.

Thus,

−τ(x+i−1)M̂h(xi−1) = τ(x−

i )M̂h(xi).

We know that M̂h = constant on Eh. If this constant is not zero then the above equationimplies that

τ(x−i ) = −τ(x+

i−1)

which is impossible since τ(x−i ) > 0 and −τ(x+

i−1) ≤ 0. Hence, this constant must be zero,i.e. M̂h = 0 on Eh, and hence (A.14b) holds.

Case (4): This was already proved in the proof of Case (2). �

Proof of Theorem 3.2 We proceed as we did in the proof of Theorem 3.1. So, we assume(A.7), and deduce from Lemma A.3 that Mh = θh = 0 in �h, and that V̂Vh = 0 on Eh. How-ever, since d = 0 we can not deduce Th = 0. Hence, we need to prove

Th = 0, in �h, (A.15a)

VVh = 0, in �h, (A.15b)

μh = 0, on Eh, (A.15c)

for each case below.Case (1): The proof that VVh = 0 in �h, and that μh = 0 on Eh follow as in the proof of

Case (1) of Theorem 3.1. Then (A.3) yields (A.15a).Case (2): The proof that VVh = 0 in �h follows as in Case (1). By (A.3) and (A.6) we

have

(Th, Th)�h+ 〈τ, (M̂h)

2〉∂�h= 〈T̂hn, M̂h〉∂�. (A.16)

Since αθ > 0 on ∂�, (A.8d) implies that M̂h = 0 on ∂�. Then, the above equation yields(A.15a) because τ ≥ 0 on ∂�h. Consequently, if k ≥ 1 then (2.1c) implies that M̂h = 0 on∂�h. If k = 0, then we deduce from (2.1c) that M̂h is constant on ∂�h, but since αθ > 0 on∂�, (A.8d) shows that this constant is zero.

Case (3): As in Cases (1) and (2) above, (A.15b) follows from the assumption that αT

or τ > 0 on at least one point of ∂Ij for all j . We know by (2.1d) with q = 0 that T̂h is aconstant on ∂�h. Then, for any v ∈ V k

h , upon integration by parts we get that (T̂h, v′)�h

=〈T̂h, vn〉∂�h

. Inserting this into (2.1d) we obtain

(Th − T̂h, v′)�h

= 0 for all v ∈ V kh .

Now, assuming that k ≥ 1, and taking v ∈ V kh such that v′ is equal to the constant T̂h we see

that

(Th − T̂h, T̂h)�h= 0. (A.17)

24 J Sci Comput (2010) 44: 1–37

Taking v3 = T̂h in (2.1c) we obtain

(Th, T̂h)�h= 〈M̂h, T̂hn〉∂�h

. (A.18)

Since, Mh = VVh = 0 in �h, and V̂Vh = 0 on Eh, the definition of the numerical trace T̂h,(2.2c), reduces to T̂h = Th + τM̂hn. Inserting this into (A.18) implies

〈M̂h, Thn〉∂�h+ 〈τ, (M̂h)

2〉∂�h= (Th, T̂h)�h

. (A.19)

Taking v3 = Th in (2.1c) gives 〈M̂h, Thn〉∂�h= (Th, Th)�h

. Thus, (A.19) can be written as

(Th, Th − T̂h)�h+ 〈τ, (M̂h)

2〉∂�h= 0.

Then, by (A.17) we get that

(Th − T̂h, Th − T̂h)�h+ 〈τ, M̂2

h〉∂�h= 0.

Thus,

τM̂h = 0 on ∂�h, and Th = T̂h = constant on �h.

If τ > 0 at some point of ∂�h, say x+i , then M̂h(xi) = 0. Then taking v3 in (2.1c) as the

linear function with support (xi−1, xi) such that v3(x−i ) = 1 and v3(x

+i−1) = 0, we see that

Th(1, v3)Ij = M̂h(xi)v3(x−i ) − M̂h(xi−1)v3(x

+i−1) = M̂h(xi) = 0

where we have used the fact that Th is constant on �h. Thus, Th = 0 on �h. Using this in(2.1c) implies 〈M̂h, vn〉∂�h

= 0, for all v ∈ V kh , and since k ≥ 1, we see that M̂h = 0 on Eh.

Case (4): We have already proved (A.15b). Integrating by parts in (2.1d) gives

−〈Th, vn〉∂�h+ (T ′

h, v)�h+ 〈T̂h, vn〉∂�h

= 0.

Suppose that k = 0, then T ′h = 0 and hence

〈T̂h − Th, vn〉∂�h= 0 for all v ∈ V 0

h .

By (2.2c), T̂h = Th + τM̂hn, and so

〈τM̂hn, vn〉∂�h= 0 for all v ∈ V 0

h . (A.20)

Taking v = χI1 , we see that

τ(x−1 )M̂h(x1) − τ(x+

0 )M̂h(x0) = 0.

Now, if we take

τ(x−1 ) = 0 and τ(x+

0 ) > 0

we see that M̂h(x0) = 0. Similarly, taking v = χIN in (A.20), we get that

τ(x−N)M̂h(xN) − τ(x+

N−1)M̂h(xN−1) = 0.

Thus taking

τ(x+N−1) = 0 and τ(x−

N) > 0

J Sci Comput (2010) 44: 1–37 25

we get that M̂h(xN) = 0. Thus M̂h = 0 on ∂�. Inserting this into (A.16) we see that

(Th, Th)�h+ 〈τ, (M̂h)

2〉∂�h\∂� = 0.

Since τ ≥ 0, we immediately deduce that Th = 0 in �h. Using this in (2.1c) we obtain

〈M̂h, vn〉∂�h= 0 for all v ∈ V 0

h

and hence M̂h is identically constant on Eh. Since M̂h = 0 on ∂� we deduce that M̂h = 0on Eh. This finishes the proof. �

It remains to prove Lemmas A.1 and A.2.

Proof of Lemma A.1 To prove the first identity we begin with taking v1 = −Th in (2.1a) andv2 = Mh in (2.1b), and adding the resulting equations to obtain

E = −(θh,M′h)�h

+ 〈θ̂h,Mhn〉∂�h+ (VVh, T

′h)�h

− 〈V̂Vh, Thn〉∂�h+ (θh, Th)�h

where

E := (Mh/EI,Mh)�h+ d2 (Th/GA,Th)�h

.

Integrating by parts on the term (VVh, T′h)�h

, and using (2.1d) with v4 = VVh, we can rewritethe last equation as

E − 〈θ̂h,Mhn〉∂�h+ 〈T̂h, VVhn〉∂�h

− 〈Th, VVhn〉∂�h

= −(θh,M′h)�h

+ (q, VVh)�h− 〈V̂Vh, Thn〉∂�h

+ (θh, Th)�h.

Integrating by parts on the term (θh,M′h)�h

, and using (2.1c) with v3 = θh, the last equationcan be written as

E − 〈(θ̂h − θh),Mhn〉∂�h+ 〈(T̂h − Th), VVhn〉∂�h

= (q, VVh)�h+ 〈M̂h, θhn〉∂�h

− 〈V̂Vh, Thn〉∂�h.

Adding to both sides the quantity

〈θ̂h − θh, M̂hn〉∂�h− 〈T̂h − Th, V̂Vhn〉∂�h

we get

E − 〈(θ̂h − θh), (Mh − M̂h)n〉∂�h+ 〈(T̂h − Th), (VVh − V̂Vh)n〉∂�h

= (q, VVh)�h+ 〈θ̂hn, M̂h〉∂�h

− 〈T̂hn, V̂Vh〉∂�h

= (q, VVh)�h+ 〈θ̂hn, M̂h〉∂� − 〈T̂hn, V̂Vh〉∂�.

In the last step we used the fact that

〈θ̂hn, M̂h〉∂�h= 〈θ̂hn, M̂h〉∂� and 〈T̂hn, V̂Vh〉∂�h

= 〈T̂hn, V̂Vh〉∂�

since �θ̂h� = �T̂h� = 0 on E ◦h by (2.1e) and (2.1f), respectively. This finishes the proof of

the first identity, (A.1), in Lemma A.1.

26 J Sci Comput (2010) 44: 1–37

To prove the second identity we begin with taking v1 = θh in (2.1a) to obtain

d2(Th/GA,θh)�h= (θh, θh)�h

+ (VVh, θ′h)�h

− 〈V̂Vh, θhn〉∂�h.

Taking v2 = VVh in (2.1b) implies

−(Mh/EI, VVh)�h= (θh, VV

′h)�h

− 〈θ̂h, VVhn〉∂�h.

Adding these two equations we get

F = (θh, θh)�h+ (VVh, θ

′h)�h

− 〈V̂Vh, θhn〉∂�h+ (θh, VV

′h)�h

− 〈θ̂h, VVhn〉∂�h

where

F := d2(Th/GA,θh)�h− (Mh/EI, VVh)�h

.

Integrating by parts on the term (VVh, θ′h)�h

we get

F = (θh, θh)�h+ 〈VVh, θhn〉∂�h

− 〈V̂Vh, θhn〉∂�h− 〈θ̂h, VVhn〉∂�h

= (θh, θh)�h+ 〈θh, (VVh − V̂Vh)n〉∂�h

− 〈θ̂h, VVhn〉∂�h

= (θh, θh)�h− 〈(θ̂h − θh), (VVh − V̂Vh)n〉∂�h

− 〈θ̂h, V̂Vhn〉∂�h

= (θh, θh)�h− 〈(θ̂h − θh), (VVh − V̂Vh)n〉∂�h

− 〈θ̂h, V̂Vhn〉∂�.

Thus,

(θh, θh)�h− 〈(θ̂h − θh), (VVh − V̂Vh)n〉∂�h

= F + 〈θ̂h, V̂Vhn〉∂�

which is (A.2).To prove the third identity we take v3 = Th in (2.1c), v4 = Mh in (2.1d), and add the

resulting equations to obtain

(Th, Th)�h+ (q,Mh)�h

= −(Mh,T′h)�h

− (Th,M′h)�h

+ 〈M̂h, Thn〉∂�h+ 〈T̂h,Mhn〉∂�h

.

Integrating by parts on the term (Mh,T′h)�h

, and carrying out some simple algebraic manip-ulations we get

(Th, Th)�h= −(q,Mh)�h

− 〈Th,Mhn〉∂�h+ 〈T̂h,Mhn〉∂�h

+ 〈M̂h, Thn〉∂�h

= −(q,Mh)�h+ 〈(T̂h − Th), (Mh − M̂h)n〉∂�h

+ 〈T̂hn, M̂h〉∂�h

= −(q,Mh)�h+ 〈(T̂h − Th), (Mh − M̂h)n〉∂�h

+ 〈T̂hn, M̂h〉∂�,

where in the last step we have used (2.1f). Hence

(Th, Th)�h− 〈(T̂h − Th), (Mh − M̂h)n〉∂�h

= −(q,Mh)�h+ 〈T̂hn, M̂h〉∂�

which is the desired identity, (A.3).This finishes the proof of Lemma A.1. �

J Sci Comput (2010) 44: 1–37 27

Proof of Lemma A.2 These identities follow simply from the definition of the numericaltraces θ̂h and T̂h given by (2.2b) and (2.2c). Indeed, (2.2c) implies

〈T̂h − Th, (VVh − V̂Vh)n〉∂�h= 〈−τ, (Mh − M̂h)(VVh − V̂Vh)〉∂�h

+ 〈αT , (VVh − V̂Vh)2〉∂�h

.

Similarly, (2.2b) gives

−〈θ̂h − θh, (Mh − M̂h)n〉∂�h= 〈τ, (VVh − V̂Vh)(Mh − M̂h)〉∂�h

+ 〈αθ , (Mh − M̂h)2〉∂�h

.

Adding these equations together we get

−〈θ̂h − θh, (Mh − M̂h)n〉∂�h+ 〈T̂h − Th, (VVh − V̂Vh)n〉∂�h

= 〈αθ , (Mh − M̂h)2〉∂�h

+ 〈αT , (VVh − V̂Vh)2〉∂�h

which is (A.4). Similarly, using (2.2b) we get (A.5), and using (2.2c) we get (A.6).This finishes the proof of Lemma A.2. �

Appendix B: Proof of the Characterization Result of Theorem 4.1

In this appendix we prove the characterization result of Theorem 4.1. The proof will bebased on the following auxiliary lemmas. Our first auxiliary result is a simple implicationof the definition of the local solvers. Indeed, adding each of the equations (4.1), (4.3), and(4.5) over K ∈ �h, we immediately obtain the following lemma.

Lemma B.1 For any (ω,μ) ∈ [L2(Eh)]2, and (v1, v2, v3, v4) ∈ [V kh ]4, we have

− (Wω,v′1)�h

+ 〈ω,v1 n〉∂�h= (�ω,v1)�h

− d2 (Tω/GA,v1)�h, (B.1a)

− (�ω,v′2)�h

+ 〈1, (�̂ω)v2 n〉∂�h= (Mω/EI, v2)�h

, (B.1b)

− (Mω,v′3)�h

= (Tω,v3)�h, (B.1c)

− (Tω,v′4)�h

+ 〈1, (̂Tω)v4 n〉∂�h= 0; (B.1d)

− (Wμ,v′1)�h

= (�μ,v1)�h− d2 (Tμ/GA,v1)�h

, (B.2a)

− (�μ,v′2)�h

+ 〈1, (�̂μ)v2 n〉∂�h= (Mμ/EI, v2)�h

, (B.2b)

− (Mμ,v′3)�h

+ 〈μ,v3 n〉∂�h= (Tμ,v3)�h

, (B.2c)

− (Tμ,v′4)�h

+ 〈1, (̂Tμ)v4 n〉∂�h= 0; (B.2d)

and that

− (Wq, v′1)�h

= (�q, v1)�h− d2 (Tq/GA,v1)�h

, (B.3a)

− (�q, v′2)�h

+ 〈1, (�̂q)v2 n〉∂�h= (Mq/EI, v2)�h

, (B.3b)

− (Mq, v′3)�h

= (Tq, v3)�h, (B.3c)

− (Tq, v′4)�h

+ 〈1, (̂Tq)v4 n〉∂�h= (q, v4)�h

. (B.3d)

28 J Sci Comput (2010) 44: 1–37

The following lemma contains some elementary identities.

Lemma B.2 (Elementary identities) For any ω,μ,m in L2(Eh), and any w in L20(Eh) we

have

〈�̂ω,mn〉∂�h= (Mω/EI,Mm)�h

+ d2 (Tω/GA,Tm)�h

+ 〈ω, (Tm) n〉∂�h+ 〈1, (�̂ω − �ω)(m − Mm) n〉∂�h

+ 〈1, (̂Tm − Tm)(Wω)n〉∂�h, (B.4a)

〈�̂VVD,mn〉∂�h= 〈VVD, (Tm) n〉∂�h

− 〈1, (̂TVVD − TVVD)(Wm) n〉∂�h

+ 〈1, (�̂m − �m)(MVVD)n〉∂�h

+ 〈1, (̂Tm − Tm)(WVVD)n〉∂�h

+ 〈1, (�̂VVD − �VVD)(m − Mm) n〉∂�h, (B.4b)

〈�̂μ,mn〉∂�h= (Mμ/EI,Mm)�h

+ d2 (Tμ/GA,Tm)�h

+ 〈1, αθ (μ − Mμ)(m − Mm)〉∂�h

+ 〈1, αT WμWm〉∂�h, (B.4c)

〈�̂q,mn〉∂�h= (q,Wm)�h

− 〈1, (̂Tq − Tq)(Wm) n〉∂�h

+ 〈1, (�̂m − �m)(Mq)n〉∂�h

+ 〈1, (̂Tm − Tm)(Wq)n〉∂�h

+ 〈1, (�̂q − �q)(m − Mm) n〉∂�h, (B.4d)

and that

〈̂Tω,wn〉∂�h= − (Mω/EI,Mw)�h

− d2 (Tω/GA,Tw)�h

− 〈1, αT (ω − Wω)(w − Ww)〉∂�h

− 〈1, αθMωMw〉∂�h, (B.5a)

〈̂TVVD,wn〉∂�h= 〈VVD, (Tw) n〉∂�h

+ 〈1, (̂Tw − Tw)(WVVD)n〉∂�h

− 〈1, (�̂VVD − �VVD)(Mw) n〉∂�h

+ 〈1, (̂TVVD − TVVD)(w − Ww) n〉∂�h

+ 〈1, (�̂w − �w)(MVVD)n〉∂�h, (B.5b)

〈̂Tμ,wn〉∂�h= − (Mμ/EI,Mw)�h

− d2 (Tμ/GA,Tw)�h

+ 〈μ, (�w) n〉∂�h+ 〈1, (�̂w − �w)(Mμ)n〉∂�h

+ 〈1, (̂Tμ − Tμ)(w − Ww) n〉∂�h, (B.5c)

〈̂Tq,wn〉∂�h= (q,Ww)�h

+ 〈1, (̂Tw − Tw)(Wq)n〉∂�h

− 〈1, (�̂q − �q)(Mw) n〉∂�h

+ 〈1, (̂Tq − Tq)(w − Ww) n〉∂�h

+ 〈1, (�̂w − �w)(Mq)n〉∂�h. (B.5d)

J Sci Comput (2010) 44: 1–37 29

We leave the proof of this lemma to the end of this appendix.The next lemma provides the explicit solution of the local solver defined by (4.1).

Lemma B.3 Suppose that the conditions of one of the cases in Theorem 3.1 (Theorem 3.2,if d = 0) are satisfied. Let k ≥ 1, and consider an arbitrary element K = (xL, xR). Then thesolution of the local solver (4.1) is given by

(Wω)(x) = ωR − ωL

hK

(x − xL) + ωL, (B.6a)

(�ω)(x) = ωR − ωL

hK

, (B.6b)

(Mω)(x) = 0, (B.6c)

(Tω)(x) = 0, (B.6d)

for any x ∈ K , where ωL = ω(xL), ωR = ω(xR), and hK = xR − xL.

Proof The fact that the local solver (4.1) with the numerical traces (4.2) defines a uniquesolution is a direct corollary of Theorem 3.1 (Theorem 3.2, if d = 0). The result now followsdirectly from substituting the solution given by (B.6) into (4.1), and verifying that it satisfiesthe equations. �

A direct implication of (B.6a) is that,

w − Ww = 0 on ∂�h. (B.7)

Hence, by the definition of the numerical traces, (4.2), and (B.6c) and (B.6d), we have

�̂ω − �ω = 0, on ∂�h, (B.8a)

T̂ω = 0 on ∂�h. (B.8b)

Using the results of Lemmas B.3 and B.1 in the identities given by Lemma B.2 we getour next auxiliary lemma.

Lemma B.4 Suppose that k ≥ 1, then for any ω,μ,m in L2(Eh), and any w in L20(Eh) we

have

(i) 〈�̂ω,mn〉∂�h= (�ω,Tm)�h

,(ii) 〈�̂q,mn〉∂�h

= (q,Wm)�h,

(iii) 〈̂Tω,wn〉∂�h= 0,

(iv) 〈̂Tμ,wn〉∂�h= (�w,Tμ)�h

,(v) 〈̂Tq,wn〉∂�h

= (q,Ww)�h.

We leave the proof of this lemma to the end of this section.We are now ready to prove Theorem 4.1.

Proof of Theorem 4.1 Using the identity (i) in Lemma B.4 with ω = ωh we get

〈�̂ωh,mn〉∂�h= (�ωh,Tm) = bh(ωh,m). (B.9)

30 J Sci Comput (2010) 44: 1–37

Using the same identity with ω = VVD we get

〈�̂VVD,mn〉∂�h= (�VVD,Tm). (B.10)

Taking μ = μh in (B.4c) gives

〈�̂μh,mn〉∂�h= ah(ωh,m). (B.11)

Combining (B.9), (B.10), (B.11), and the identity (ii) of Lemma B.4 yields

〈(�̂ωh + �̂VVD + �̂μh + �̂q),mn〉∂�h= ah(μh,m) + bh(ωh,m) − ��

h (m) + 〈θN ,mn〉∂�.

Hence, (2.1e) implies (4.9a).Similarly, using the identities (iii), (iv), and (v) of Lemma B.4 we get

〈̂Tωh,wn〉∂�h= 0,

〈̂TVVD,wn〉∂�h= 0,

〈̂Tμh,wn〉∂�h= (�w,Tμh)�h

= bh(w,μh),

〈̂Tq,wn〉∂�h= (q,Ww)�h

.

Hence,

〈(̂Tωh + T̂VVD + T̂μh + T̂q),wn〉∂�h= bh(w,ωh) − �T

h (w).

The result, (4.9b), now follows from (2.1f). This completes the proof of the theorem. �

It remains to prove Lemma B.2 and Lemma B.4.

Proof of Lemma B.2 We only give a proof of the identities (B.4a) and (B.4b), the proofs ofthe remaining ones are very similar. To prove (B.4a) we begin by writing

〈�̂ω,mn〉∂�h= 〈�̂ω − �ω,mn〉∂�h

+ 〈�ω,mn〉∂�h. (B.12)

Taking μ = m and v3 = �ω in (B.2c) we get

〈�ω,mn〉∂�h= (Mm, (�ω)′)�h

+ (Tm,�ω)�h. (B.13)

By a simple integration by parts

(Mm, (�ω)′)�h= 〈1, (Mm)(�ω)n〉∂�h

− ((Mm)′,�ω)�h. (B.14)

Using (B.1a) with v1 = Tm yields

(�ω,Tm)�h= −(Wω, (Tm)′)�h

+ 〈ω, (Tm) n〉∂�h+ d2(Tω/GA,Tm)�h

. (B.15)

Inserting (B.14) and (B.15) into (B.13) we obtain

〈�ω,mn〉∂�h= 〈1, (Mm)(�ω)n〉∂�h

− ((Mm)′,�ω)�h

− (Wω, (Tm)′)�h+ 〈ω, (Tm) n〉∂�h

+ d2(Tω/GA,Tm)�h. (B.16)

J Sci Comput (2010) 44: 1–37 31

Integrating by parts on the term (Wω, (Tm)′)�h, and taking v2 = Mm in (B.1b) we obtain

−(Wω, (Tm)′)�h= −〈1, (Wω)(Tm) n〉∂�h

+ ((Wω)′,Tm)�h,

−((Mm)′,�ω)�h= −〈1, (�̂ω)(Mm) n〉∂�h

+ (Mω/EI,Mm)�h.

Inserting these into (B.16) implies

〈�ω,mn〉∂�h= (Mω/EI,Mm)�h

+ d2(Tω/GA,Tm)�h

+ 〈ω, (Tm) n〉∂�h− 〈1, (Wω)(Tm) n〉∂�h

+ ((Wω)′,Tm)�h− 〈1, (�̂ω − �ω)(Mm) n〉∂�h

.

Taking μ = m and v4 = Wω in (B.2d), and inserting the resulting identity into the lastequation above we get that

〈�ω,mn〉∂�h= (Mω/EI,Mm)�h

+ d2(Tω/GA,Tm)�h

+ 〈ω, (Tm) n〉∂�h− 〈1, (�̂ω − �ω)(Mm) n〉∂�h

+ 〈1, (̂Tm − Tm)(Wω)n〉∂�h.

Combining this with (B.12) we reach at (B.4a).To prove (B.4b) we need to further work on the first two terms on the right-hand side of

(B.4a). To do that, we start with taking μ = m and v1 = Tω in (B.2a) to obtain

d2(Tm/GA,Tω)�h= (�m,Tω)�h

+ (Wm, (Tω)′)�h. (B.17)

Integrating by parts we obtain

(Wm, (Tω)′)�h= 〈1, (Wm)(Tω)n〉∂�h

− ((Wm)′,Tω)�h. (B.18)

Taking v3 = �m in (B.1c) gives

(�m,Tω)�h= −(Mω, (�m)′)�h

. (B.19)

Using (B.18) and (B.19) in (B.17) we get

d2(Tm/GA,Tω)�h= −(Mω, (�m)′)�h

+ 〈1, (Wm)(Tω)n〉∂�h− ((Wm)′,Tω)�h

.

(B.20)

Integrating by parts on the term (Mω, (�m)′)�hand taking v4 = Wm in (B.1d) we obtain

−(Mω, (�m)′)�h= −〈1, (Mω)(�m) n〉∂�h

+ ((Mω)′,�m)�h, (B.21)

and

−((Wm)′,Tω)�h= −〈1, (̂Tω)(Wm) n〉∂�h

. (B.22)

Inserting (B.21) and (B.22) into (B.20) implies

d2(Tm/GA,Tω)�h= −〈1, (Mω)(�m) n〉∂�h

+ ((Mω)′,�m)�h

− 〈1, (̂Tω − Tω)(Wm) n〉∂�h. (B.23)

32 J Sci Comput (2010) 44: 1–37

Taking μ = m and v2 = Mω in (B.2b) we get

((Mω)′,�m)�h= 〈1, (�̂m)(Mω)n〉∂�h

− (Mm/EI,Mω)�h.

Inserting this into (B.23) we get

(Mm/EI,Mω)�h+ d2(Tm/GA,Tω)�h

= −〈1, (̂Tω − Tω)(Wm) n〉∂�h+ 〈1, (�̂m − �m)(Mω)n〉∂�h

.

Combining this with (B.4a), and setting ω = VVD in the resulting expression, we finallyobtain (B.4b). �

Proof of Lemma B.4 (i) Taking v1 = Tm in (B.1), we obtain

−(Wω, (Tm)′)�h+ 〈ω, (Tm) n〉∂�h

= (�ω,Tm)�h− d2(Tω/GA,Tm)�h

.

Similarly, taking μ = m and v4 = Wω in (B.2), we get

−(Tm, (Wω)′)�h+ 〈1, (̂Tm)(Wω)n〉∂�h

= 0.

Combining these two equations, and carrying out some simple algebraic manipulations, weobtain

〈ω, (Tm) n〉∂�h+ 〈1, (̂Tm − Tm)(Wω)n〉∂�h

= (�ω,Tm)�h− d2(Tω/GA,Tm)�h

.

Using this in (B.4a) we get

〈�̂ω,mn〉∂�h= (�ω,Tm)�h

+ (Mω/EI,Mm)�h

+ 〈1, (�̂ω − �ω)(m − Mm) n〉∂�h.

The identity (i) now follows from (B.6c) and (B.8a).(ii) This follows from inserting the definition of the numerical traces, (4.4), and (4.6),

into the elementary identity (B.4d), and carrying out some simplifications.(iii) This follows trivially from (B.8b).(iv) Using (B.6c), (B.6d), (B.7), and (B.8), in (B.5c) we get

〈̂Tμ,wn〉∂�h= 〈μ, (�w)n〉∂�h

.

Taking v3 = �w in (B.2c) we get

−(Mμ, (�w)′)�h+ 〈μ, (�w) n〉∂�h

= (Tμ,�w)�h.

The identity (iv) now follows from the last two equations once we note that (�w)′ = 0 by(B.6b).

(v) This is a simple result of inserting (B.6c), (B.6d), (B.7), and (B.8) into (B.5d). �

J Sci Comput (2010) 44: 1–37 33

Appendix C: Proof of Theorem 5.1

In this appendix we prove Theorem 5.1. The key to the proof is the fact that it is possibleto compute the numerical traces M̂h and V̂Vh of the HDG method explicitly in terms of theaverages and the jumps of the approximate solution (Th,Mh, θh, VVh). We state and provethis result next.

In this appendix, we will make a slight abuse of notation and denote both the DG solutionand the HDG solution as (Th,Mh, θh, VVh), i.e. we will drop the superscript “H ” in the HDGsolution. We can do this because the solutions will be identical a posteriori.

Proposition C.1 Consider the HDG method defined by the weak formulation (2.1) and theformulas (2.2) for the numerical traces. Then at every interior node x ∈ E ◦

h

M̂h = {{Mh}} + 1

2D

[(�τn��τ � + �αT n��αθ �)�Mh�

+ 2�τ �αT�VVh� − 2�αT n��θh� − 2�τn��Th�

], (C.1)

and

V̂Vh = {{VVh}} + 1

2D

[(�τn��τ � + �αθn��αT �)�VVh�

− 2�τ �αθ�Mh� + 2�αθn��Th� − 2�τn��θh�

]. (C.2)

Before proving this proposition, let us collect some elementary algebraic identities.

Lemma C.2 For functions κ , α, and ϕ which are possibly double valued on E ◦h we have

�κn��αϕn� − �αn��κϕn� = −�ϕ��κ�α, (C.3a)

�αϕn� = �αn�{{ϕ}} + 1

2�α��ϕ�. (C.3b)

Proof The first identity is proved as follows

�κn��αϕn� − �αn��τϕn� = (κ+ + κ−)(α+ϕ+ + α−ϕ−)

− (α+ + α−)(κ+ϕ+ + κ−ϕ−)

= ϕ+(κ+α+ + κ−α+ − α+κ+ − α−κ+)

− ϕ−(−κ+α− − κ−α− + α+κ− + α−κ−)

= (ϕ+ − ϕ−)(α+κ− − α−κ+)

= −�ϕ� · �κ�α.

The second identity can be proved by expanding the right-hand side and showing that it isequal to the left-hand side after simplifications. �

Proof of Proposition C.1 By the conservativity conditions (2.1e) and (2.1f), we have that�T̂h� = �θ̂h� = 0 at every interior node. Hence, by the definition of the numerical traces,

34 J Sci Comput (2010) 44: 1–37

(2.2b) and (2.2c), we have

0 = �Th� − �τMhn� + �τn�M̂h + �αT VVhn� − �αT n�V̂Vh,

0 = �θh� − �αθMhn� + �αθn�M̂h − �τVVhn� + �τn�V̂Vh

which is a simple linear system for M̂h and V̂Vh. Inverting this system yields

M̂h = 1

D

[�τn��τMhn� + �αT n��αθMhn�

− �τn��αT VVhn� + �αT n��τVVhn�

− �αT n��θh� − �τn��Th�], (C.4)

V̂Vh = 1

D

[�τn��τVVhn� + �αθn��αT VVhn�

+ �τn��αθMhn� − �αθn��τMhn�

− �τn��θh� + �αθn��Th�]. (C.5)

Using the identity (C.3a) with κ = τ , α = αT , and ϕ = VVh, we obtain

−�τn��αT VVhn� + �αT n��τVVhn� = �VVh� · �τ �αT.

Using the same identity with κ = τ , α = αθ , and ϕ = Mh, we get

�τn��αθMhn� − �αθn��τMhn� = −�Mh� · �τ �αθ.

Inserting these into (C.4) and (C.5) we get

M̂h = 1

D

[�τn��τMhn� + �αT n��αθMhn�

+ �τ �αT− �αT n��θh� − �τn��Th�

], (C.6)

V̂Vh = 1

D

[�τn��τVVhn� + �αθn��αT VVhn�

− �τ �αθ− �τn��θh� + �αθn��Th�

]. (C.7)

Using the identity (C.3b) with α = τ and ϕ = Mh we get

�τMhn� = �τn�{{Mh}} + 1

2�τ ��Mh�.

Similarly, α = αθ and ϕ = Mh we get

�αθMhn� = �αθn�{{Mh}} + 1

2�αθ ��Mh�.

Hence,

�τn��τMhn� + �αT n��αθMhn�

J Sci Comput (2010) 44: 1–37 35

= {{Mh}}(�τn�2 + �αT n��αθn�) + 1

2

(�τn��τ � + �αT n��αθ �

)�Mh�.

Upon dividing both sides by D = �τn�2 + �αT n��αθn� we get

�τn��τMhn� + �αT n��αθMhn�

D= {{Mh}} + �τn��τ � + �αT n��αθ �

2D�Mh�, (C.8)

and similarly,

�τn��τVVhn� + �αθn��αT VVhn�

D= {{VVh}} + �τn��τ � + �αθn��αT �

2D�VVh�. (C.9)

Inserting (C.8) and (C.9) into (C.6) and (C.7), respectively, we reach at (C.1) and (C.2). �

Proof of Theorem 5.1 Since both the HDG method and the DG method are defined by thesame weak formulation, their solutions are identical if and only if their numerical traces,(T̂h, M̂h, θ̂h, V̂Vh), are identical at every node of the mesh. We will prove that this is thecase if we pick the coefficients Cij as prescribed in the theorem. We proceed in five steps,in the first four steps we give the proof for the interior nodes, namely, we will derive theidentities (5.4)–(5.7), and in the last one we give a proof for the boundary nodes, namely,the identity (5.8).

Step 1: Proof of (5.4).

Inserting (5.4) into (5.1a), and comparing the resulting expression with (C.2) shows that thenumerical trace V̂Vh of the DG method and that of the HDG method are identical.


Inserting the expressions for M̂h and V̂Vh provided, respectively, by (C.1) and (C.2) into(2.2b) we obtain

θ̂+h = θ+

h + α+θ

[

− 1

2�Mh� − �τn��τ � + �αT n��αθ �

2D�Mh�

− �τ �αT

D�VVh� + �αT n�

D�θh� + �τn�

D�Th�

]

+ τ+[

− 1

2�VVh� − �τn��τ � + �αθn��αT �

2D�VVh�

+ �τ �αθ

D�Mh� − �αθn�

D�Th� + �τn�

D�θh�

]

.

After a recollection of terms we can rewrite this equation as

θ̂+h = −

(α+

θ �τ �αT

D+ τ+

(1

2+ �τn��τ � + �αθn��αT �

2D

))

�VVh�

+ θ+h + α+

θ �αT n� + τ+�τn�

D�θh�

− α+θ

(1

2+ �τn��τ � + �αT n��αθ �

2D− τ+�τ �αθ

D

)

�Mh�

36 J Sci Comput (2010) 44: 1–37

+ α+θ �τn� − τ+�αθn�

D�Th�.

Comparing this with (5.1b), and carrying out some simple algebraic manipulations we seethat the two numerical traces are identical if we pick C2j as prescribed by (5.5).


Inserting (5.6) into (5.1c), and comparing the resulting expression with (C.1) shows that thenumerical trace M̂h of the DG method and that of the HDG method are identical.


We proceed as we did in the proof of Step 2. Inserting the expressions for M̂h and V̂Vh

provided, respectively, by (C.1) and (C.2) into (2.2c), and recollecting terms, we obtain

T̂ +h = −

(τ+�τ �αT

D− α+

T

(1

2+ �τn��τ � + �αθn��αT �

2D

))

�VVh�

+ τ+�αT n� − α+T �τn�

D�θh�

−(

α+T �τ �αθ

D+ τ+

(1

2+ �τn��τ � + �αT n��αθ �

2D

))

�Mh�

+ T +h + τ+�τn� + α+

T �αθn�

D�Th�.

Comparing this with (5.1d), and carrying out some simple algebraic manipulations we seethat the two numerical traces are identical if we pick C4j as prescribed by (5.7).


We only prove this for x = 0, the proof for x = 1 is similar.The conservativity condition (2.1e) implies θ̂h(0) = θN(0). Thus, the numerical trace

(2.2b) at x = 0 can be written as

θN(0) = θh(0+) + αθ(0

+)(Mh(0+) − M̂h(0)) + τ(0+)(VVh(0

+) − VVD(0)).

Using the assumption αθ (0+) > 0 we get

M̂h(0) = Mh(0+) + 1

αθ (0+)

[− τ(0+)(VVD(0) − VVh(0

+)) − (θN(0) − θh(0+))

]. (C.10)

Comparing this with (5.2c) we see that the numerical trace M̂h(0) of the DG method isidentical to that of the HDG method if we set C31(0) and C32(0) as prescribed by (5.8).

Similarly, using (C.10) in (2.2c) we get

T̂h(0+) = Th(0

+) + τ 2(0+) + αT (0+)αθ (0+)

αθ (0+)(VVD(0) − VVh(0

+))

+ τ(0+)

αθ (0+)(θN(0) − θh(0

+)).

Comparing this with (5.2d) we see that the numerical trace T̂h(0) of the DG method isidentical to that of the HDG method if we set C41(0) and C42(0) as prescribed by (5.8).

This completes the proof of Theorem 5.1. �

J Sci Comput (2010) 44: 1–37 37

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2. Celiker, F., Cockburn, B.: Element-by-element post-processing of discontinuous Galerkin methods forTimoshenko beams. J. Sci. Comput. 27(1–3), 177–187 (2006)

3. Celiker, F., Cockburn, B., Güzey, S., Kanapady, R., Soon, S.-C., Stolarski, H.K., Tamma, K.K.: Discontin-uous Galerkin methods for Timoshenko beams. In: Numerical Mathematics and Advanced Applications,ENUMATH 2003, pp. 221–231. Springer, Berlin (2003)

4. Celiker, F., Cockburn, B., Stolarski, H.K.: Locking-free optimal discontinuous Galerkin methods for Tim-oshenko beams. SIAM J. Numer. Anal. 44(6), 2297–2325 (2006)

5. Cockburn, B., Dong, B., Guzmán, J.: A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77(264), 2008 (1887–1916)

6. Cockburn, B., Dong, B., Guzmán, J.: A hybridizable and superconvergent discontinuous Galerkin methodfor biharmonic problems. J. Sci. Comput. 40, 141–187 (2009)

7. Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed,and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009)

8. Cockburn, B., Guzmán, J., Wang, H.: Superconvergent discontinuous Galerkin methods for second-orderelliptic problems. Math. Comput. 78, 1–24 (2009)

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