superconvergence of the direct discontinuous galerkin method for convection...

Superconvergence of the Direct DiscontinuousGalerkin Method for Convection-DiffusionEquationsWaixiang Cao,1 Hailiang Liu,2 Zhimin Zhang1,3

1Division of Applied and Computational Mathematics, Beijing Computational ScienceResearch Center, Beijing 100193, China

2Department of Mathematics, Iowa State University, Ames, Iowa 500113Department of Mathematics, Wayne State University, Detroit, Michigan 48202

Received 12 October 2015; accepted 6 July 2016Published online in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/num.22087

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG)method for one-dimensional linear convection-diffusion equations. We prove, under some suitable choiceof numerical fluxes and initial discretization, a 2k-th and (k + 2) -th order superconvergence rate of theDDG approximation at nodes and Lobatto points, respectively, and a (k + 1) -th order of the derivativeapproximation at Gauss points, where k is the polynomial degree. Moreover, we also prove that the DDGsolution is superconvergent with an order k + 2 to a particular projection of the exact solution. Numericalexperiments are presented to validate the theoretical results. © 2016 Wiley Periodicals, Inc. Numer MethodsPartial Differential Eq 000: 000–000, 2016

Keywords: convection-diffusion equations; direct discontinuous Galerkin methods; superconvergence

I. INTRODUCTION

In this paper, we study the superconvergence of the direct discontinuous Galerkin (DDG) methodfor the one-dimensional linear convection-diffusion equation

∂tu + ∂xf (u) = ∂2x u, (x, t) ∈ [a, b] × [0, T ],

u(x, 0) = u0(x), x ∈ [a, b], (1.1)

Correspondence to: Waixiang Cao, Beijing Computational Science Research Center, Beijing 100094, China (e-mail:[email protected])Contract grant sponsor: China Postdoctoral Science Foundation; contract grant number: 2015M570026Contract grant sponsor: National Natural Science Foundation of China (NSFC); contract grant numbers: 11501026,11471031, and 91430216Contract grant sponsor: US National Science Foundation (NSF); contract grant number: DMS-1419040Contract grant sponsor: US National Science Foundation (NSF); contract grant number: DMS-1312636 and RNMS(KI-Net) 1107291 (H. Liu)

© 2016 Wiley Periodicals, Inc.

2 CAO, LIU, AND ZHANG

where the linear flux f = αu. For simplicity, we only consider the case α ≥ 0 and the periodicboundary condition.

For equations containing higher-order spatial derivatives, such as the convection-diffusionequation (1.1), the discontinuous Galerkin (DG) method [1–6] cannot be directly applied, dueto the discontinuous solution space at the element interfaces, which is not regular enough tohandle higher derivatives. Several DG methods have been suggested in the literature to solvethis problem, including the local discontinuous Galerkin (LDG) method [7], the interior penalty(IP) method [8–10], and the method introduced by Baumann-Oden [11, 12], and the direct DGdiscretization methods (see, e.g., [13–15]). In 2009, Liu and Yan [16] introduced a new direct DGmethod for convection-diffusion problems. The basic idea of this method is to directly force theweak formulation of the PDE into the DG method, without introducing any auxiliary variablesor rewriting the original equation into a first-order system. It was also discussed in this paperhow a careful choice of numerical fluxes can be made to ensure stability and accuracy of theDDG method. Later, Liu et al. studied a set of more effective choices of parameters in the DDGnumerical fluxes [17, 18], and proved an optimal L2 error estimates for the DDG methods solvingconvection-diffusion problems in [19].

The key feature of DDG methods lies in its special choice of the numerical flux for the solutionderivative at cell interfaces. This gives DDG methods extra flexibility and advantage over theIPDG method [20] and the LDG method [7]. The DDG method is the only known DG methodwhich can satisfy the maximum principle for Fokker-Planck equations up to third order [21], yetboth IPDG method and LDG method can be proved to satisfy maximum principle with only upto second order of accuracy even for diffusion without drift (see e.g. [22]). The additional para-meter in the DDG numerical flux also makes it possible for the DG method for Fokker–Plancktype equations to satisfy some entropy dissipation law at the discrete level, see, e.g. [23, 24].The superconvergence result obtained for a nontrivial β1 presented in this paper provides anotherexample of advantages of the DDG method.

The main purpose of this paper is to study and reveal the superconvergence phenomenon ofthe DDG method for the convection-diffusion equation (1.1). To the best of our knowledge, therewas no any superconvergence result of the DDG method for these problems in the literature. Ourresults include the superconvergence properties of the DDG approximation at nodal and Lobattopoints, and the derivative approximation at Gauss points, as well as the supercloseness betweenthe DDG solution and a particular projection of the exact solution. To be more precise, we provethat, under a careful choice of the parameters in the DDG numerical fluxes, the DDG solution issuperconvergent at all Gauss (derivative approximation) and Lobatto points, with an order k + 1and k + 2, respectively; and the error between the DDG solution and the Gauss–Lobatto projectionof the exact solution achieves (k + 2) -th order. We further prove a 2k-th order superconvergencerate of the DDG approximation at nodes. As we may recall, all these superconvergence results arethe same as the counterpart finite element methods (see. e.g., [25–27]). Moreover, the influenceof the choice of the parameters in the numerical fluxes on the superconvergence is also discussed.

The analysis is based on the correction function idea, which is motivated from its successfulapplication to the FEM and finite volume methods (FVM) for Poisson equations [28, 29], and theDG or LDG methods for hyperbolic and parabolic equations [30–32]. The essence of the correc-tion idea is to construct a specially designed function which vanishes or is of high order at somespecial points. With the correction function, we first construct a special interpolation function ofthe exact solution, and then prove the interpolation function is superconvergent to the numericalsolution, with the highest order 2k. Finally the supercloseness between the interpolation functionand the numerical solution gives the desired superconvergence results at some special points:Gauss, Lobatto, and nodal points.

Numerical Methods for Partial Differential Equations DOI 10.1002/num

SUPERCONVERGENCE OF THE DDG METHOD 3

The main difficulty in the superconvergence analysis lies in the construction of the correctionfunction (or the special interpolation function) and the choice of the parameters in the numericalfluxes. The correction function here is greatly different from those in [30–32], due to the dif-ferent model problems and numerical schemes. The existence of the convection-diffusion termf (u) and the special choice of numerical fluxes make the construction of the correction functionmore sophisticated. It is also different from the correction functions of FEM or FVM for ellipticequations in [28, 29] because of the time-dependent feature. Furthermore, as the choice of theparameters in the numerical fluxes influences the superconvergence directly (a feature not sharedby those methods in [28–32]), how a careful choice of the parameters in the numerical fluxes toensure some desired superconvergence results (including superconvergence at Gauss and Lobattopoints) is also a technical question needed to handle.

The rest of the paper is organized as follows. In section 2, we recall the DDG schemes andthe basic stability results for linear convection-diffusion equations, following [19]. Sections 3 and4 are the main body of the paper, where superconvergence results for zero flux and linear fluxare proved separately, with suitable initial discretization and careful choice of numerical fluxes.Finally, some numerical examples are presented in section 5 to support our theoretical findings.

Throughout this paper, we adopt standard notations for Sobolev spaces such as Wm,p(D) onsubdomain D ⊂ � equipped with the norm ‖·‖m,p,D and seminorm |·|m,p,D . When D = �, we omitthe index D; and if p = 2, we set Wm,p(D) = Hm(D), ‖ · ‖m,p,D = ‖ · ‖m,D , and | · |m,p,D = | · |m,D .We use the notation A � B to indicate that A can be bounded by B multiplied by a constantindependent of the mesh size h. A � B stands for A � B and B � A.

II. DDG SCHEMES

To discretize the weak formulation, we divide the interval � = [a, b] with a mesh: a = x 12

< x 32

<

· · · < xN+ 1

2= b, and the mesh size h = hj = x

j+ 12

− xj− 1

2, and the interval Ij = [x

j− 12, x

j+ 12].

We denote by v+ and v− the right and left limits of any function v, and define

[v] = v+ − v−, {v} = v+ + v−

2.

Define the k-degree discontinuous finite element space

Vh ={vh : vh|Ij ∈ Pk(Ij ), j ∈ ZN

},

where Pk(Ij ) denotes the set of all polynomials of degree no more than k on Ij , and Zr ={1, 2, . . . , r} for any positive integer r.

The DDG method for (1.1) is to find uh ∈ Vh such that for any vh ∈ Vh

(∂tuh, vh)j = (f (uh) − ∂xuh, ∂xvh)j + ((−f (uh) + ∂xuh)vh + (uh − {uh})∂xvh)|∂Ij. (2.1)

Here f (uh) and ∂xuh are numerical fluxes, and

(ϕ, v)j =∫

Ij

ϕvdx, v|∂Ij= v

(x−

j+ 12

)− v

(x+

j− 12

), ∀ϕ, v ∈ L2.



Following [19], we take the numerical fluxes f (uh) = αu−h and

∂xuh = β0

h[uh] + {∂xuh} + β1h[∂2

x uh] (2.2)

with βi , i = 0, 1 satisfying the following stability condition

β0 ≥ �(β1),

where

�(β1) = supν∈Pk−1[−1,1]

2(ν(1) − 2β1∂sν(1))2∫ 1−1 ν2(s)ds

.

Summing up all j and using the periodic boundary condition, we have from (2.1)

N∑j=1

(∂tuh, vh)j +N∑

j=1

(−f (uh) + ∂xuh, ∂xvh)j +N∑

j=1

((−f (uh) + ∂xuh)[vh] + [uh] {∂xvh})|j+ 12

= 0,

or equivalently,

(∂tuh, vh) + A(uh, vh) − F(uh, vh) = 0, ∀vh ∈ Vh, (2.3)

where (uh, vh) = ∑N

j=1 (uh, vh)j , and

A(uh, vh) =N∑

j=1

(∂xuh, ∂xvh)j +N∑

j=1

(∂xuh[vh] + {∂xvh} [uh])|j+ 12, (2.4)

F(uh, vh) =N∑

j=1

(f (uh), ∂xvh)j +N∑

j=1

f (uh)[vh]|j+ 12. (2.5)

For any function vh ∈ Vh satisfying the periodic boundary condition, we have (see [19])

F(vh, vh) ≤ 0, A(vh, vh) ≥ γ ‖vh‖2E ,

where γ ∈ (0, 1) is some positive constant and

‖vh‖2E =

N∑j=1

∫Ij

|∂xvh|2dx +N∑

j=1

β0

h[vh]2|

j+ 12. (2.6)

Then

1

2

d

dt‖vh‖2

0 + γ ‖vh‖2E ≤ (∂tvh, vh) + A(vh, vh) − F(vh, vh), ∀vh ∈ Vh. (2.7)

III. SUPERCONVERGENCE FOR ZERO FLUX f = 0

Before the study of superconvergence, we begin by introducing some notations, two importantprojections Ph and Ih, and the supercloseness between Ihv and Phv for any smooth function v,which will be frequently used in our later superconvergence analysis.



A. Preliminaries

First, for a smooth function v, we define a global projection Phv ∈ Vh of v by

(Phv, ϕh)j = (v, ϕh)j , ∀ϕh ∈ Pk−2(Ij ), j ∈ ZN , (3.1)

∂x(Phv) := β0h−1[Phv] + {∂x(Phv)} + β1h[∂2

x (Phv)]|j+ 1

2= ∂xv(x

j+ 12), (3.2)

{Phv} |j+ 1

2= v(x

j+ 12), j ∈ ZN . (3.3)

Note that we use the periodic boundary condition at xN+ 1

2, and Ph only needs to satisfy the condi-

tions (3.2)–(3.3) when k = 1. It is shown in [19] that the global projection Ph is uniquely defined,and

‖v − Phv‖0 � hk+1‖v‖k+1. (3.4)

Second, denote by Lm the Legendre polynomial of degree m, and {φm}∞0 the series of Lobatto

polynomials, on the interval [−1, 1]. That is, φ0 = 1−s

2 , φ1 = s+12 , and

φm+1(s) =∫ s

−1Lm(s ′)ds ′, m ≥ 1, s ∈ [−1, 1].

The orthogonal property of Legendre polynomials gives

φm(−1) = φm(1) = 0, m ≥ 2. (3.5)

Moreover, there holds the identity

φm = 1

2m − 1(Lm − Lm−2), m ≥ 2.

Therefore, we have for all m, r ≥ 2,∫ 1

−1φmφrds = 1

(2m − 1)(2r − 1)

∫ 1

−1(Lm − Lm−2)(Lr − Lr−2)ds = 0, if m − r �= 0, ±2.

(3.6)

Let Lj ,m and φj ,m be the Legendre and Lobatto polynomials of degree m on the interval Ij ,respectively. That is,

Lj ,m(x) = Lm(s), φj ,m(x) = φm(s), s =2x − x

j− 12

− xj+ 1

2

h, m ≥ 0.

For any smooth function v, we have the following expansion in each element Ij ,

v(x) =∞∑

m=0

vj ,mφj ,m(x), ∀x ∈ Ij ,

where

vj ,0 = v(x+

j− 12

), vj ,1 = v

(x−

j+ 12

), vj ,m = 2m − 1

2

∫Ij

∂xvLj ,m−1dx, m ≥ 2. (3.7)



It has been proved in [29] that

|vj ,m+1| � hi+1− 1

p ‖v‖i+1,p,Ij , 0 ≤ i ≤ m. (3.8)

Now we define the Gauss–Lobatto projection of v by

Ihv|Ij :=k∑

m=0

vj ,mφj ,m(x). (3.9)

By (3.5)–(3.6) and a scaling from [−1, 1] to Ij , we have

φj ,m

(x+

j− 12

) = φj ,m

(x−

j+ 12

) = 0, φj ,m+1 ⊥ Pm−2(Ij ), ∀m ≥ 2,

which yields

(v − Ihv)(x−

j+ 12

) = (v − Ihv)(x+

j− 12

) = 0, (v − Ihv) ⊥ Pk−2.

Moreover, since

∂x(v − Ihv)(x) = 2

h

∞∑m=k+1

vj ,mLj ,m−1(x), x ∈ Ij , (3.10)

we obtain from the orthogonality of the Legendre polynomials

∂x(v − Ihv) ⊥ Pk−1. (3.11)

To estimate the error Ihv−Phv for any smooth function v, we need the following important result.

Lemma 3.1. Suppose M is an N × N block circulant matrix with the first row [CBO . . . O]and the last row [BO . . . OC], where O is an 2 × 2 zero matrix, and both C and B are 2 × 2nonsingular matrix satisfying det(C ± B) �= 0. Then

|ρ(M−1)| ≤ c,

where ρ(M) denotes the spectral radius of M, and c is some bounded constant independent of N.

Proof. For any vector or matrix X, we first denote by ‖X‖ the L2 norm of X. Let ωj =ei

2πjN , j = 0, . . . , N − 1 be N roots of 1 with i2 = −1, and �j =

(ωj 00 ωj

). It is easy to verify

from the properties of the circulant matrix

Mζj = (C + B�j)ζj , ζj = (I , �j , �2j , . . . , �N−1

j )T

,

where I is the 2 × 2 identity matrix. By denoting

ϒ =(

ζ0

‖ζ0‖ ,ζ1

‖ζ1‖ , . . . ,ζN−1

‖ζN−1‖)

, � = diag(C + B�0, C + B�1, . . . , C + B�N−1),



we have M = ϒ�ϒ−1. By a direct calculation,

(ζl)T ζj =

(∑N−1m=0 ei

(j−l)2πmN 0

0∑N−1

m=0 ei(j−l)2πm

N

)=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(1 0

0 1

), ifj = l,(

0 0

0 0

), ifj �= l.

Then ϒ is an orthogonal matrix, which yields

‖ϒ‖ = ‖ϒ−1‖ = 1.

On the other hand, since det(C ± B) �= 0, both M and � are nonsingular matrices. Then eachC + B�j is invertible and thus,

|ρ(M−1)| = ‖M−1‖ ≤ ‖ϒ‖‖�−1‖‖ϒ−1‖ � maxj

det(C + B�j)−1 � 1.

Then the desired result follows.

Now we are ready to show the supercloseness between Ihv and Phv.

Lemma 3.2. For any function v ∈ Hk+2, if β1 = 12k(k+1)

, then

‖Phv − Ihv‖0 � hk+2‖v‖k+2. (3.12)

Proof. Suppose Phv − Ihv has the following formula in Ij ,

(Phv − Ihv)(x) =k∑

m=0

vj ,mLj ,m(x), x ∈ Ij .

Recalling the definition of Ph and the properties of Ih, we have

(Phv − Ihv, ϕh)j = 0, ϕh ∈ Pk−2(Ij ), {Phv − Ihv} |j+ 1

2= 0,

( ∂x(Phv) − ∂x(Ihv))|j+ 1

2= (∂xv − ∂x(Ihv))|

j+ 12,

which yields vj ,m = 0, m ≤ k − 2, and

k∑m=k−1

(Lm(1)vj ,m + Lm(−1)vj+1,m) = 0,

k∑m=k−1

(g0(m)vj ,m + g1(m)vj+1,m) = h(∂xv − ∂x(Ihv))|j+ 1

2= bj ,

where

g0(m) = −β0Lm(1) + L′m(1) − 4β1L

′′m(1), g1(m) = β0Lm(−1) + L′

m(−1) + 4β1L′′m(1).

(3.13)



Let M be an N × N block circulant matrix with the first row [CBO . . . O] and the last row[BO . . . OC], where O is an 2 × 2 zero matrix, and

C =(

Lk−1(1) Lk(1)

g0(k − 1) g0(k)

), B =

(Lk−1(−1) Lk(−1)

g1(k − 1) g1(k)

).

Then{vj

}N

j=1with vj = (vj ,k−1, vj ,k) satisfies the linear system

MX = b, X = (v1, . . . , vN )T

, b = (0, b1, 0, b2, . . . , 0, bN)T . (3.14)

It is proved in [19] that det(C ± B) �= 0 when β0 > �(β1). Then the conclusion in Lemma 3.1gives

N∑j=1

k∑m=k−1

v2j ,m � ‖X‖2 � ‖M−1‖2‖b‖2 �

N∑j=1

b2j .

Now we estimate the term bj , j ∈ ZN . Noticing that v − Ihv is continuous across the point xj+ 1

2,

we have

bj = h(∂xv − ∂x(Ihv))|j+ 1

2= h

({∂x(v − Ihv)} |

j+ 12

+ β1h[∂2x (v − Ihv)]|

j+ 12

).

Plugging (3.18) into the above identity and using the scaling from [−1, 1] to Ij , j ∈ ZN yields

bj =∞∑

m=k+1

(vj ,m(Lm−1(1) − 4β1L′m−1(1)) + vj+1,m(Lm−1(−1) + 4β1L

′m−1(−1)))

=∞∑

m=k+1

(vj ,m + (−1)m−1vj+1,m)(1 − 2β1m(m − 1)),

where the coefficient vj ,m is the same as in (3.15), and in the last step, we have used the followingidentity

Lm−1(±1) = (±1)m−1, L′m−1(±1) = 1

2(±1)mm(m − 1).

Consequently, if β1 = 12k(k+1)

, we have

bj =∞∑

m=k+2

(vj ,m + (−1)m−1vj+1,m)(1 − 2β1m(m − 1)),

which yields, together with (3.16),

|bj | � hk+2− 12 (‖v‖k+2,Ij + ‖v‖k+2,Ij+1). (3.16)

Here we use the notation IN+1 = I1. Note that

‖Phv − Ihv‖20 � h

N∑j=1

k∑m=k−1

v2j ,m � h

N∑j=1

b2j .

Then the desired result (3.20) follows.



We end this subsection with an integral projection D−1x , which is defined as, for any function v,

D−1x v|Ij :=

∫ x

xj− 1

2

vdx.

Apparently,

‖D−1x v‖0 � h‖v‖0. (3.15)

Moreover, noticing that

D−1x (v − Ihv)(x) =

∞∑m=k+1

vj ,m

∫ x

xj− 1

2

φj ,mdx =∞∑

m=k+1

vj ,m

2m − 1

∫ x

xj− 1

2

(Lj ,m − Lj ,m−2)dx, x ∈ Ij ,

we have all for k ≥ 2,

D−1x (v − Ihv)

(x−

j+ 12

) = D−1x (v − Ihv)

(x+

j− 12

) = 0, ∀j ∈ ZN . (3.16)

Similarly, since

(v − Phv) ⊥ P0, k ≥ 2,

we have

(v − Phv)|Ij =∞∑

m=1

vj ,mLj ,m(x), vj ,m = 2m + 1

h(v − Phv, Lj ,m)

j.

Then

D−1x (v − Phv)(x−

j+ 12) = D−1

x (v − Phv)(x+j− 1

2) = 0, ∀j ∈ ZN , k ≥ 2. (3.17)

The identities (3.16) and (3.17) indicate that both D−1x (v− Ihv) and D−1

x (v−Phv) are continuouson the whole domain � when k ≥ 2.

B. Superconvergence at Gauss and Lobatto Points

To study the superconvergence properties of the DDG solution uh at Gauss and Lobatto points, wefirst analyze the supercloseness of uh toward the Gauss–Lobatto projection Ihu of the exact solu-tion u, and then use the supercloseness result between uh and Ihu to obtain the superconvergenceat Gauss and Lobatto points. Our analysis is along this line.

Define

H 1h =

{v : v|Ij ∈ H 1(Ij ), j ∈ ZN

},

and for all ϕ, v ∈ H 1h , let

a(ϕ, v) = A(ϕ, v) + (ϕ, v)



and

‖|v‖|E = ‖v‖E + ‖v‖0,

where ‖ · ‖E is defined in (2.6). For any vh ∈ Vh, the inverse inequality holds. Then

{∂xvh}2|j+ 1

2� ‖∂xvh‖2

0,∞,Ij+ ‖∂xvh‖2

0,∞,Ij+1� h−1(‖∂xvh‖2

0,Ij+ ‖∂xvh‖2

0,Ij+1), j ∈ ZN .

Similarly, we get

h[∂2x vh]2|

j+ 12

� h‖∂2x vh‖2

0,∞,Ij+ h‖∂2

x vh‖20,∞,Ij+1

� h−1(‖∂xvh‖20,Ij

+ ‖∂xvh‖20,Ij+1

).

Consequently,

h

N∑j=1

({∂xvh} + β1h[∂2x vh])2|

j+ 12

�N∑

j=1

∫Ij

|∂xvh|2dx.

Recalling the definition of A(·, ·) in (2.4), we have for all ϕh, vh ∈ Vh,

A(ϕh, vh) � ‖ϕh‖E

(N∑

j=1

∫Ij

|∂xvh|2dx + h

N∑j=1

{∂xvh}2|j+ 1

2+ h−1

N∑j=1

[vh]2|j+ 1

2

) 12

� ‖ϕh‖E‖vh‖E .

Then

a(vh, vh) ≥ γ ‖|vh‖|2E , a(ϕh, vh) � ‖|ϕh‖|E‖|vh‖|E , ∀ϕh, vh ∈ Vh. (3.18)

By the Lax–Milgram lemma, there exists a unique function wu ∈ Vh such that

a(wu, vh) = −(∂t (u − Phu), vh), ∀vh ∈ Vh. (3.19)

Choosing vh = wu in the above equation and using the coercivity of a(·, ·), we obtain for any k ≥ 2,

γ ‖|wu‖|2E ≤ |(∂t (u − Phu), wu)| = |(D−1x ∂t (u − Phu), ∂xwu)|

� hk+2‖∂tu‖k+1‖|wu‖|E ,

where we have used the integration by parts and (3.17) in the second step, and (3.4), (3.15) in thelast step. Noticing ut = uxx , then

‖|wu‖|E � hk+2‖ut‖k+1 � hk+2‖u‖k+3. (3.20)

Similarly, taking time derivative on both sides of (3.19) and choosing v = ∂twu yields

‖|∂twu‖|E � hk+2‖utt‖k+1 � hk+2‖u‖k+5. (3.21)

We have the following superconvergence results for uh − Ihu and uh − Phu.



Theorem 3.3. Let u ∈ Hk+5 and uh be the solution of (1.1) and (2.1), respectively. Suppose theinitial value uh(x, 0) is chosen such that

‖uh(·, 0) − Ihu0‖0 � hk+2‖u0‖k+3. (3.22)

Then there holds for all k ≥ 2 and t > 0,

‖(uh − Phu)(·, t)‖0 + ‖(uh − Ihu)(·, t)‖0 � hk+2 supτ∈[0,t]

‖u(·, τ)‖k+5. (3.23)

Denote

e = u − uh, ξ = uh − Phu + wu, η = u − Phu + wu,

where wu is the solution of (3.19). Noticing that the exact solution u also satisfies (2.3), we obtain,by choosing vh = ξ in (2.7) and using the orthogonality (∂te, vh) + A(e, vh) = 0, vh ∈ Vh,

1

2

d

dt‖ξ‖2

0 ≤ |(∂tη, ξ) + A(η, ξ)|= |(∂t (u − Phu), ξ) + (∂twu, ξ) + A(wu, ξ) + A(u − Phu, ξ)|= |(∂twu, ξ) − (wu, ξ) + A(u − Phu, ξ)|.

Here in the last step, we have used the identity (3.19). By integration by parts, there holds for allϕ ∈ H 1

h and vh ∈ Vh

A(ϕ, vh) = −(ϕ, ∂2x vh) +

N∑j=1

(−[ϕ∂xvh] + ∂xϕ[vh] + {∂xvh} [ϕ])|j+ 1

2

= −(ϕ, ∂2x vh) +

N∑j=1

(∂xϕ[vh] − {ϕ} [∂xvh])|j+ 12. (3.24)

By the properties of Ph, we obtain

A(u − Phu, vh) = 0, ∀vh ∈ Vh. (3.25)

Consequently,

1

2

d

dt‖ξ‖2

0 ≤ |(∂twu, ξ) − (wu, ξ)| � (‖∂twu‖0 + ‖wu‖0)‖ξ‖0.

In light of (3.20)–(3.21), we have

d

dt‖ξ‖0 � hk+2‖u‖k+5,

which yields

‖ξ(·, t)‖0 � ‖ξ(·, 0)‖0 + hk+2

∫ t

0||u(·, τ)‖k+5dτ � ‖ξ(·, 0)‖0 + T hk+2 sup

τ∈[0,t]‖u(·, τ)‖k+5.



Due to the special choice of the initial solution, (3.12) and (3.20), we get

‖ξ(·, 0)‖0 � hk+2‖u0‖k+2 + ‖wu(·, 0)‖0 � hk+2‖u0‖k+3,

and thus,

‖ξ(·, t)‖0 � hk+2 supτ∈[0,t]

‖u(·, τ)‖k+5. (3.26)

Then the desired result (3.23) follows from (3.12) and (3.20).

Remark 3.4. The result of Theorem 3.3 indicates that the DDG solution uh is superclose withan order k + 2 to the global projection Phu and the Gauss–Lobatto projection Ihu of the exactsolution.

We denote by gj ,m, (j , m) ∈ ZN × Zk and lj ,m, (j , m) ∈ ZN × Zk+1 the Gauss points of degreek and Lobatto points of degree k + 1 on the interval Ij , respectively. That is, gj ,m and lj ,m are sep-arately zeros of Lj ,k and φj ,k+1. As a direct consequence of Theorem 3.3, we have the followingsuperconvergence results of the derivative approximation at Gauss points and the function valueapproximation at Lobatto points.

Corollary 3.5. Suppose all the conditions of Theorem 3.3 hold. Then for any fixed t ∈ (0, T ],

eg =(

1

Nk

N∑j=1

k∑i=1

∂x(u − uh)2(gj ,i , t)

) 12

� hk+1 supτ∈[0,t]

‖u(·, τ)‖k+5, (3.27)

el =(

1

N(k + 1)

N∑j=1

k+1∑i=1

(u − uh)2(lj ,i , t)

) 12

� hk+2 supτ∈[0,t]

‖u(·, τ)‖k+5. (3.28)

Proof. In each element Ij , j ∈ ZN , note that

(v − Ihv)(lj ,i ) =∞∑

m=k+2

vj ,mφj ,m(lj ,i ), i ∈ Zk+1,

(v − Ihv)x(gj ,i ) = 2

h

∞∑m=k+2

vj ,mLj ,m−1(gj ,i ), i ∈ Zk ,

where vj ,m is defined by (3.7). Therefore, for any v ∈ Hk+2, we have from (3.8)

(v − Ihv)(li,j ) � hk+2− 12 ‖v‖k+2,Ij , ∂x(v − Ihv)(gi,j ) � hk+1− 1

2 ‖v‖k+2,Ij ,

which yields

(1

N(k + 1)

N∑j=1

k+1∑i=1

(v − Ihv)2(lj ,i )

) 12

� hk+2‖v‖k+2,



(1

Nk

N∑j=1

k∑i=1

∂x(v − Ihv)2(gj ,i )

) 12

� hk+1‖v‖k+2.

On the other hand, since Ihu − uh ∈ Vh, we have from the inverse inequality

(1

N(k + 1)

N∑j=1

k+1∑i=1

(Ihu − uh)2(lj ,i )

) 12

� ‖Ihu − uh‖0,

(1

Nk

N∑j=1

k∑i=1

∂x(Ihu − uh)2(gj ,i )

) 12

� h−1‖Ihu − uh‖0.

Then the desired results (3.27)–(3.28) follow from (3.23) and the triangle inequality.

C. Superconvergence at Nodes

As a direct consequence of (3.23), we have the following superconvergence result at nodes.

(1

N

N∑j=1

{u − uh}2|j+ 1

2

) 12

=(

1

N

N∑j=1

{Ihu − uh}2|j+ 1

2

) 12

� ‖Ihu − uh‖0 � hk+2 supτ∈[0,t]

‖u(·, τ)‖k+5.

However, our numerical results show that the superconvergence rate at nodes can reach 2k whenk ≥ 3, which is better than the order k + 2 we provide above. Therefore, new analysis tools areneeded to prove the 2k-th superconvergence order for k ≥ 3.

We will adopt the idea of correction function in our superconvergence analysis. That is, weconstruct a specially designed function w ∈ Vh such that w vanishes or is of high order at nodes(to guarantee that the error is small enough to be compatible with the superconvergence errorestimate at nodes) and

‖uh − uI‖0 = ‖uh − Phu + w‖0 ≤ h2kc(u),

where c(u) is a constant dependent on the exact solution u (our later superconvergence analysisshows that c(u) can be bounded by supτ∈[0,T ]‖u(·, τ)‖2k+2 ). In light of (2.7) and (3.25),

1

2

d

dt‖uh − uI‖2

0 + γ ‖uh − uI‖2E ≤ (∂t (uh − uI ), uh − uI ) + A(uh − uI , uh − uI )

= (∂t (u − uI ), uh − uI ) + A(u − uI , uh − uI )

= (∂t (u − uI ), uh − uI ) + A(w, uh − uI ). (3.29)

Therefore, to achieve our superconvergence goal, we need to construct a function w ∈ Vh suchthat the right hand side in (3.29) is of high order. To be more precise, we shall construct a functionw ∈ Vh satisfying

(∂t (u − uI ), uh − uI ) + A(w, uh − uI ) ≤ h2kc1(u)(‖uh − uI‖E + ‖uh − uI‖0).



Here again, c1(u) is dependent on some norm of the exact solution u.To design w, we first denote w0 = u − Phu and define a serial of functions wi , 1 ≤ i ≤

(k − 1)/2� as follows.

(wi , ∂2x vh) = (∂twi−1, vh), ∀vh ∈ Pk(Ij )\P1(Ij ), (3.30)

∂xwi := β0h−1[wi] + {∂xwi} + β1h[∂2

xwi]|j+ 12

= 0, (3.31)

{wi} |j+ 1

2= 0. (3.32)

Here k� denotes the maximal integer no more than k.Due to the existence of the global projection Ph, the function wi , 1 ≤ i ≤ (k − 1)/2�

defined in (3.30)–(3.32) is uniquely determined. Moreover, we have the following estimate forwi , 1 ≤ i ≤ (k − 1)/2�.

Lemma 3.6. For any k ≥ 3, suppose wi , 1 ≤ i ≤ (k − 1)/2� is defined by (3.30)–(3.32). Then

wi |Ij :=k∑

m=k−2i−1

cij ,mLj ,m, (3.33)

where Lj ,−1 = 0 and cij ,m are some bounded constants. Furthermore, if ∂r

t u ∈ Hk+1+2i , r =0, 1, 1 ≤ i ≤ (k − 1)/2�, there holds

‖∂rt wi‖0 � hk+1+2i‖u‖k+1+2(i+r). (3.34)

Proof. We will show (3.33) by induction. First, we suppose in each element Ij

wi |Ij :=k∑

m=0

cij ,mLj ,m.

For any vh ∈ Pk\P1, since ∂2x vh ∈ Pk−2, by choosing ∂2

x vh = Lj ,m, m ∈ Zk−2 in (3.30), weimmediately obtain

cij ,m = 2m + 1

h

∫Ij

∂twi−1(D−1x D−1

x Lj ,m)dx, m ≤ k − 2. (3.35)

Noticing that D−1x D−1

x Lj ,m ∈ Pm+2(Ij ) and (u − Phu) ⊥ Pk−2, then

c1j ,m = 2m + 1

h

∫Ij

∂t (u − Phu)(D−1x D−1

x Lj ,m)dx = 0, ∀m ≤ k − 4,

which indicates that (3.33) is valid for i = 1. Suppose (3.33) holds for all i, 1 ≤ i ≤ (k − 1)/2�−1.Then

wi ⊥ Pk−2i−2.

In light of (3.35), we have

ci+1j ,m = 2m + 1

h

∫Ij

∂twi(D−1x D−1

x Lj ,m)dx = 0, ∀m < k − 2i − 3.

Consequently, (3.33) is also valid for i + 1. Then the proof of (3.33) is complete.



We next estimate the coefficients cij ,m. By (3.35) and (3.15), we have

|cij ,m| ≤ h−1‖∂twi−1‖0,Ij ‖D−1

x D−1x Lj ,m‖0,Ij � h

32 ‖∂twi−1‖0,Ij , m ≤ k − 2.

To estimate cij ,m, m = k − 1, k, we obtain from (3.31)–(3.32)

k∑m=k−1

(Lm(1)cij ,m + Lm(−1)ci

j+1,m) = −k−2∑

m=k−2i−1

(Lm(1)cij ,m + Lm(−1)ci

j+1,m),

k∑m=k−1

(g0(m)cij ,m + g1(m)ci

j+1,m) = −k−2∑

m=k−2i−1

(g0(m)cij ,m + g1(m)ci

j+1,m),

where g0, g1 are the same as in (3.13). By the same argument as in Lemma 3.2, we have

N∑j=1

k∑m=k−1

(cij ,m)

2 �N∑

j=1

k−2∑m=k−2i−1

(cij ,m)

2 � h3‖∂twi−1‖20.

Consequently,

‖wi‖0 �(

N∑j=1

k∑m=k−2i−1

h(cij ,m)

2

) 12

� h2‖∂twi−1‖0, 1 ≤ i ≤ (k − 1)/2� .

Taking time derivative on both sides of (3.30)–(3.32), the three identities still hold. In other words,we can replace wi by ∂twi in (3.30)–(3.32). Similarly, (3.30)–(3.32) are still valid if we take ∂m

t wi

instead of wi , where m ≥ 1. Then following the same arguments as what we did for wi , we get

‖∂mt wi‖0 � h2‖∂m+1

t wi−1‖0, ∀m ≥ 1, 1 ≤ i ≤ (k − 1)/2� .

By recursion formula, there holds for all 1 ≤ i ≤ (k − 1)/2� and r = 0, 1

‖∂rt wi‖0 � h2i‖∂r+i

t w0‖0.


‖∂mt w0‖0 = ‖∂m

t (u − Phu)‖0 � hk+1‖∂mt u‖k+1 � hk+1‖u‖k+1+2m, m ≥ 1.

Then (3.34) follows. This finishes our proof.

Now we define the special correction function w as

w = (k−1)/2�∑

i=1

wi . (3.36)

Theorem 3.7. Let u and uh be the solution of (1.1) and (2.1), respectively, and uI = Phu − w

be the special interpolation function with w defined by (3.36), (3.30)–(3.32). For all k ≥ 3, ifu ∈ H 2k+2, then

‖(uh − uI )(·, t)‖0 � ‖(uh − uI )(·, 0)‖0 + h2k supτ∈[0,t]

‖u(·, τ)‖2k+2, ∀t > 0. (3.37)



Proof. By (3.33) and the orthogonality of Phu, we have

wi ⊥ P1, ∀0 ≤ i ≤ (k − 1)/2� − 1.

Then for any function vh ∈ P1,

(∂twi , vh) = 0 = (wi+1, ∂2x vh), ∀0 ≤ i ≤ (k − 1)/2� − 1,

which gives, together with (3.30)

(wi+1, ∂2x vh) = (∂twi , vh), ∀vh ∈ Vh, 1 ≤ i ≤ (k − 1)/2� − 1.

Consequently, for any vh ∈ Vh, we have from (3.24) and (3.31)–(3.32),

(∂t (u − Phu), vh) + A(w, vh) + (∂tw, vh) = (∂twr , vh), if k = 2r + 1,

(∂t (u − Phu), vh) + A(w, vh) + (∂tw, vh) = (∂twr−1, vh)

= −(D−1x ∂twr−1, ∂xvh), if k = 2r .

Here in the last step for the even k = 2r , we have used integration by parts, and the fact thatwr−1 ⊥ P1, which yields

D−1x wr−1(x

−j+ 1

2) = D−1

x wr−1(x+j− 1

2) = 0.

In light of (3.15) and (3.34), we have for all k ≥ 3

|(∂t (u − Phu), vh) + A(w, vh) + (∂tw, vh)| ≤ C1h2k‖u‖2k+2(‖vh‖0 + ‖vh‖E)

≤ C2h4k‖u‖2

2k+2 + C3‖vh‖20 + γ

2‖vh‖2

E ,

where Ci , i ≤ 3 are some bounded constants independent of the mesh size h. Plugging the aboveinequality into (3.29) yields

d

dt‖uI − uh‖2

0 � h4k‖u‖22k+2 + ‖uI − uh‖2

0,

and thus,

‖(uI − uh)(·, t)‖20 � ‖(uI − uh)(·, 0)‖2

0 + h4k supτ∈[0,t]

‖u(·, τ)‖22k+2 +

∫ t

0||(uI − uh)(·, τ)‖2

0dτ .

By the Gronwall inequality (see, e.g., [33], p.9), we have

‖(uI − uh)(·, t)‖20 � (1 + et )

(‖(uI − uh)(·, 0)‖2

0 + h4k supτ∈[0,t]

‖u(·, τ)‖22k+2

).

Then the desired result (3.37) follows.



Theorem 3.8. Let u ∈ H 2k+2 be the solution of (1.1), and uh be the solution of (2.1). SupposeuI = Phu−w with w defined by (3.36), (3.30)–(3.32), and the initial value uh(x, 0) is taken suchthat

‖(uh − uI )(·, 0)‖0 � h2k‖u0‖2k+2. (3.38)

Then there holds for all k ≥ 3 and t ∈ (0, T ]

en =(

1

N

N∑j=1

(u − {uh})2(xj+ 1

2, t)

) 12

� h2k supτ∈[0,t]

‖u(·, τ)‖2k+2. (3.39)

Proof. First, for any fixed t ∈ (0, T ], by the special choice of the initial value and (3.37), wehave

‖(uI − uh)(·, t)‖0 � h2k supτ∈[0,t]

‖u(·, τ)‖2k+2.

On the other hand, it is easy to obtain from (3.32) that {w(·, t)} |j+ 1

2= 0. Then

u(xj+ 1

2, t) = {u(·, t)} |

j+ 12

= {(Phu − w)(·, t)} |j+ 1

2= {uI (·, t)} |

j+ 12.

Since uh − uI ∈ Vh, the inverse inequality holds, and thus(1

N

N∑j=1

({uI } − {uh})2(xj+ 1

2, t)

) 12

�(

1

N

N∑j=1

||(uI − uh)(·, t)‖20,∞,Ij

) 12

� ‖(uI − uh)(·, t)‖0.

Then

en � ‖(uI − uh)(·, t)‖0 � h2k supτ∈[0,t]

‖u(·, τ)‖2k+2.

This finishes our proof.

Remark 3.9. We would like to point out that, based on the superconvergence results in Theorem3.3, Corollary 3.5 and Theorem 3.8, the choice of β1 in (2.1) has influence on the superconvergenceof the DDG solution uh at Gauss and Lobatto points, as well as the supercloseness between uh andIhu. However, it does not affect the superconvergence rate of uh at nodes. We will demonstratethese points in our numerical experiments.

Remark 3.10. In our superconvergence analysis in Corollary 3.5 and Theorem 3.8, the initialdiscretizations (3.22) and (3.38) are of great significance. To obtain the k + 2-th order supercon-vergence rate at Gauss and Lobatto points, the initial value uh(x, 0) = Ihu0 or Phu0 is a validchoice. However, to achieve the 2k goal at nodal points, it is indicated from Theorem 3.8 that theinitial error should also reach the same superconvergence rate, which imposes a stronger conditionon the initial discretization. A natural way of initial discretization to obtain (3.46) is to chooseuh(x, 0) = uI (x, 0), or uh(x, 0) = uI (x, 0), where

uI ={

uI if k = 2r

uI − wr if k = 2r + 1(3.40)



The latter is a valid choice due to the fact that

‖wr(·, 0)‖0 � h2k‖u0‖2k , when k = 2r + 1.

D. Initial Discretization

In this subsection, we would like to demonstrate how to implement the initial discretization as itplays an important role in our superconvergence analysis. If we take the initial data uh(x, 0) = Ihu0

or uh(x, 0) = Phu0, then we can obtain uh(x, 0) by only using the information of the initialvalue u0 and (3.1)-(3.3) and (3.9). While if the initial data is chosen as uh(x, 0) = uI (x, 0) oruh(x, 0) = uI (x, 0), the initial discretization is somewhat complicated. We next show how tocalculate uI (x, 0) only using the information of u0. The same technique can be applied to uI .Noticing that

∂rt u|t=0 = ∂2r

x u0, 0 ≤ r ≤ (k − 1)/2� ,

then

∂rt w0|t=0 = ∂r

t (u − Phu)|t=0 = ∂2rx u0 − Ph(∂

2rx u0).

On the other hand, taking time derivative on both sides of (3.30)–(3.32), we can calculate the term∂r

t wi+1|t=0 from the information of ∂r+1t wi |t=0. Now we divide the initial discretization into the

following steps:

1. Compute Ph(∂2rx u0), 0 ≤ r ≤ (k − 1)/2� by (3.1)–(3.3) and set ∂r

t w0 = ∂2rx u0−Ph(∂

2rx u0).

2. Calculate ∂rt wi+1, i ≤ (k − 1)/2� , 0 ≤ r ≤ k − i by (3.30)–(3.32) with wi+1 replaced by

∂rt wi+1 and ∂twi replaced by ∂r+1

t wi .3. Figure out uh(x, 0) = Phu0 − ∑ (k−1)/2�

i=1 wi .

IV. SUPERCONVERGENCE FOR LINEAR FLUX f = αU

The superconvergence analysis for the convection-diffusion equation with f = αu is similar tothat for the zero flux f = 0. To simplify our proof, we only consider the case in which α is aconstant in the rest of this section.

We first introduce another global projection Qh. For a given smooth function v, the projectionQhv ∈ Vh is defined by

(Qhv, ϕh)j = (v, ϕh)j , ∀ϕh ∈ Pk−2(Ij ), j ∈ ZN , (4.1)

( ∂x(Qhv) − f (Qhv))|j+ 1

2= ∂xv(x

j+ 12) − (αv)(x

j+ 12), (4.2)

{Qhv} |j+ 1

2= v(x

j+ 12). (4.3)

The uniqueness and existence of the projection Qh have been proved in [19]. Moreover, thereholds

‖u − Qhu‖0 � hn‖u‖n, 0 ≤ n ≤ k + 1. (4.4)

We have the following superconvergence result of Qhv − Ihv.



Lemma 4.1. For any function v ∈ Hk+2, if β1 = 12k(k+1)

, then

‖Qhv − Ihv‖0 � hk+2‖v‖k+2. (4.5)

Here we omit the proof since it is similar to that for zero flux, the only difference lies in amodification of g0

g0(m) = g0(m) − αh,

where g0 is given in (3.13).By the definition of Qh, we have

A(u − Qhu, vh) − F(u − Qhu, vh) = −(α(u − Qhu), ∂xvh), ∀vh ∈ Vh. (4.6)

We modify the correction function wi ∈ Vh as follows. For all k ≥ 2, wi , 1 ≤ i ≤ k − 1 is thefunction satisfying

(wi , ∂2x vh)j

= (∂twi−1, vh)j − (αwi−1, ∂xvh)j , ∀vh ∈ Pk(Ij )\P1(Ij ), (4.7)

(∂xwi − f (wi))|j+ 12

= 0, (4.8)

{wi} |j+ 1

2= 0, j ∈ ZN . (4.9)

Here w0 = u − Qhu. Note that the definition of wi is the same as that of the global projectionQh, except the right hand side. Therefore, wi , 1 ≤ i ≤ k − 1 defined in (4.7)–(4.9) is uniquelydetermined.

Lemma 4.2. Suppose wi ∈ Vh, i ∈ Zk−1 is defined by (4.7)–(4.9), and in each elementIj , j ∈ ZN ,

wi |Ij :=k∑

m=0

cij ,mLj ,m.

Then there holds for any positive integer r

(h

N∑j=1

(∂rt c

ij ,m)

2

) 12

� hl‖∂rt u‖l , 0 ≤ m ≤ k, (4.10)

where 0 ≤ l ≤ max(2k − m, k + 1 + i). Consequently,

‖∂rt wi‖0 � hk+1+i‖∂r

t u‖k+1+i . (4.11)

Proof. Since (4.11) is a direct consequence of (4.10), we only prove (4.10) in the follow-ing. We will show (4.10) by induction. First, noticing that (4.7)–(4.9) still hold by taking timederivative of r-th order, then we choose ∂2

x vh = Lj ,m, m ≤ k − 2 in (4.7) to derive

h

2m + 1∂r

t ci+1j ,m = (∂1+r

t wi , D−1x D−1

x Lj ,m)j− (α(∂r

t wi), D−1x Lj ,m)

j, m ≤ k − 2. (4.12)



Since w0 = (u − Qhu) ⊥ Pk−2 and D−1x D−1

x Lj ,m ∈ Pm+2(Ij ), D−1x Lj ,m ∈ Pm+1(Ij ), we have, by

letting i = 0 in (4.12)

∂rt c

1j ,m = 0, ∀m = 0, . . . , k − 4.

On the other hand, note that

‖D−1x D−1

x Lj ,m‖0,Ij � h52 , ‖D−1

x Lj ,m‖0,Ij � h32 , m ≥ 0.

By choosing i = 0, m = k − 2 in (4.12) and using the estimate (4.4) and the fact ut = uxx −αux ,we obtain for all 0 ≤ n ≤ k + 1

(h

N∑j=1

(∂rt c

1j ,k−2)

2

) 12

� h2‖∂1+rt w0‖0 + h‖∂r

t w0‖0

� hn+1(‖∂rt u‖n + ‖∂1+r

t u‖n−1) � hn+1‖∂rt u‖n+1.

Moreover, if k ≥ 3, we have (∂rt w0, D−1

x Lj ,k−3)j= 0, and thus,

|∂rt c

1j ,k−3| = 2m + 1

h|∫

Ij

∂1+rt w0(D

−1x D−1

x Lj ,k−3)dx| � h32 ‖∂1+r

t (u − Qhu)‖0,Ij ,

which yields

(h

N∑j=1

(∂rt c

1j ,k−3)

2

) 12

� h2‖∂1+rt (u − Qhu)‖0 � hn+2‖∂1+r

t u‖n � hn+2‖∂rt u‖n+2.

To estimate ∂rt c

1j ,m, m = k − 1, k, we follow the same argument as in Lemma 3.2 to obtain

(h

N∑j=1

k∑m=k−1

(∂rt c

1j ,m)

2

) 12

�(

h

N∑j=1

k−2∑m=k−3

(∂rt c

1j ,m)

2

) 12

� hn+1‖∂rt u‖n+1.

Consequently, (4.10) is valid for all r ≥ 0 and m, 0 ≤ m ≤ k in case i = 1. Now we suppose (4.10)is valid for i, 1 ≤ i ≤ k − 2 and prove that the same result holds true for i + 1. In fact, by (4.12)and the orthogonality of the Legendre polynomials,

|∂rt c

i+1j ,m | � h2

(m+2∑p=0

|∂r+1t ci

j ,p|)

+ h

(m+1∑p=0

|∂rt c

ij ,p|

), m ≤ k − 2.

For any positive integer p, p ≤ m+ 2, by choosing n′ ≤ max(2k −m− 2, k + i), we obtain fromthe induction hypothesis,

(h

N∑j=1

(∂1+rt ci

j ,p)2

) 12

≤ hn′ ‖∂1+rt u‖n′ = hn′ ‖∂r

t u‖n′+2,



which yields

h2 max0≤p≤m+2

(h

N∑j=1

(∂1+rt ci

j ,p)2

) 12

� hn′+2‖∂rt u‖n′+2 = hn‖∂r

t u‖n,

where n = n′ + 2 ≤ max(2k − m, k + 1 + (i + 1)). Similarly, we can obtain

h max0≤p≤m+1

(h

N∑j=1

(∂rt c

ij ,p)

2

) 12

� hn‖∂rt u‖n.

Then a direct calculation indicates that (4.10) is valid for i + 1 when 0 ≤ m ≤ k−2. By (4.8)–(4.9)and the same argument as in Lemma 3.2, we obtain(

h

N∑j=1

k∑m=k−1

(∂rt c

i+1j ,m)

2

) 12

�(

h

N∑j=1

k−2∑m=0

(∂rt c

i+1j ,m)

2

) 12

� hk+2+i‖∂rt u‖k+2+i .

Therefore, (4.10) also holds true for i + 1 when m = k − 1, k. Thus (4.10) is valid for i + 1. Thisfinishes our proof.

To study the superconvergence properties of uh at Gauss and Lobatto points, we define

w = w1, uI = Qhu − w = Qhu − w1.

Then a direct calculation from (2.5), (3.24), (4.8)–(4.9) yields

A(w, vh) − F(w, vh) = −(w, ∂2x vh) − (αw, ∂xvh). (4.13)

On the other hand, choosing vh = uh − uI = ξ in (2.7) and using the orthogonal property, we get

1

2

d

dt‖ξ‖2

0 + γ ‖ξ‖2E ≤ |(∂t (u − uI ), ξ) + A(u − uI , ξ) − F(u − uI , ξ)|. (4.14)

In light of (4.6)–(4.9), we have for all v ∈ Pk\P1,

(∂t (u − uI ), v) + A(u − uI , v) − F(u − uI , v) = (∂tw1, v) − (αw1, vx).

While for any v ∈ P1, noticing that ∂xv ∈ P0 and (u − Qhu) ⊥ P0, we have from (4.6)

A(u − Qhu, v) − F(u − Qhu, v) = 0,

and thus,

((u − uI )t , v) + A(u − uI , v) − F(u − uI , v) = (∂tw1, v) − (αw1, ∂xv) + (∂t (u − Qhu), v)

= (∂tw1, v) − (D−1x ∂t (u − Qhu) + αw1, ∂xv).

Here in the last step, we have used the integration by parts and the fact that

D−1x ∂t (u − Qhu)(x−

j+ 12) = D−1

x ∂t (u − Qhu)(x−j− 1

2) = 0.



Since any vh ∈ Vh can be decomposed into vh = v + v with v ∈ P1, v ∈ Pk\P1, we obtain

(∂t (u − uI ), vh) + A(u − uI , vh) − F(u − uI , vh)

= (∂tw1, vh) − (αw1, ∂xvh) − (D−1x ∂t (u − Qhu), ∂xv),

which yields, together with (4.14) and the fact that ‖v‖0 + ‖v‖0 � ‖vh‖0,

1

2

d

dt‖uh − uI‖2

0 � ‖∂tw1‖20 + ‖w1‖2

0 + ‖D−1x ∂t (u − Qhu)‖2

0 + ‖uh − uI‖2.

By (4.11) and the Gronwall inequality,

‖(uh − uI )(·, t)‖0 � ‖(uh − uI )(·, 0)‖0 + hk+2 supτ∈[0,t]

‖u(·, τ)‖k+5, ∀t > 0.

Consequently, if we choose the initial value uh(x, 0) = Ihu0(x) or uh(x, 0) = Qhu0(x), then

‖(uh − uI )(·, 0)‖0 � ‖(Ihu − Qhu)(·, 0)‖0 + ‖w1(·, 0)‖0 � ‖u0‖k+2.

Thus,

‖(uh − uI )(·, t)‖0 ≤ hk+2 supτ∈[0,t]

‖u(·, τ)‖k+5, ∀t > 0.

By following the same argument as in the case f = 0, we obtain the same superconvergence resultsat Gauss and Lobatto points for the convection-diffusion flux f = αu. That is, the results inCorollary 3.5 are valid for the convection-diffusion equation f = αu.

Now we focus on our attentions to the 2k-th superconvergence rate of the DDG approximationat nodes for k ≥ 3. We modify the correction function w as

w =k−1∑i=1

wi , (4.15)

where wi is defined by (4.7)–(4.9). In this case, (4.13) still holds. Then for all v ∈ Pk\P1,

|((u − uI )t , v) + A(u − uI , v) − F(u − uI , v)| = |(∂twk−1, v) − (αwk−1, vx)|� h2k‖∂tu‖2k(‖v‖0 + |v|1).

For any v ∈ P1(Ij ), we suppose v = vj ,0Lj ,0 + vj ,1Lj ,1. Then there holds for all i, 1 ≤ i ≤ k − 1,

(∂twi , v)j − α(wi , vx)j =N∑

j=1

(h∂tc

ij ,0vj ,0 + h

3∂tc

ij ,1vj ,1 − 2αci

j ,0vj ,1

).


(N∑

j=1

h(∂tcij ,0)

2 + h(cij ,0)

2 + h3(∂tcij ,1)

2

) 12

� h2k‖∂tu‖2k .



Then we use the Cauchy–Schwartz inequality to obtain

|(∂twi , v) − (wi , vx)| � h2k‖∂tu‖2k(‖v‖0 + |v|1), ∀v ∈ P0.

Noticing that (u − Qhu) ⊥ P1 when k ≥ 3, we get

(∂t (u − uI ), v) + A(u − uI , v) − F(u − uI , v) = (∂tw, v) − (αw, vx)

� h2k‖∂tu‖2k(‖v‖0 + |v|1).Then for all vh = v + v ∈ Vh,

|(∂t (u − uI ), vh) + A(u − uI , vh) − F(u − uI , vh)| � h2k‖∂tu‖2k(‖vh‖0 + |vh|1).Consequently, we have from (4.14), the Cauchy–Schwartz inequality, and the Gronwall inequality,

‖(uh − uI )(·, t)‖0 � ‖(uh − uI )(·, 0)‖0 + h2k supτ∈[0,t]

‖∂tu(·, τ)‖2k , ∀t > 0.

If the initial value is taken as uh(x, 0) = uI (x, 0), we immediately obtain

‖(uh − uI )(·, t)‖0 � h2k supτ∈[0,t]

‖∂tu(·, τ)‖2k � h2k supτ∈[0,t]

‖u(·, τ)‖2k+2, ∀t > 0.

Then by the same arguments as in Theorem 3.8, the superconvergence result (3.39) follows forthe convection-diffusion flux f = αu.

Remark 4.3. Following the argument as in the constant coefficient case, we can obtain the samesuperconvergence results for the variable coefficient α, where α is sufficiently smooth.

V. NUMERICAL EXAMPLES

In this section, we present numerical examples to verify our theoretical findings. To test the super-convergence phenomenon of uh at Gauss, Lobatto and nodal points, as well as the superclosenessof uh toward Ihu, we will measure in our numerical experiments the errors ‖uh − Ihu‖0 andel , eg , en (defined in Theorem 3.8 and Corollary 3.5).

In our experiments, we will use DDG scheme (2.1) for spatial discretization, and the fourth-order Runge–Kutta method for time discretization with the time step �t = 0.001h2. We obtainour meshes by equally dividing (0, 2π) into N(4, . . . , 64) subintervals.

Example 1. We consider the following equation with the periodic boundary condition:

ut = uxx , (x, t) ∈ (0, 2π) × (0, T ), u(x, 0) = sin(x).

The exact solution is

u = e−t sin(x).

We take different ways of initial discretizations and different values of (β0, β1) to test theinfluence of the initial errors and the choice of parameters in the DDG numerical fluxes (2.2) onthe superconvergence rate.



TABLE I. Errors and corresponding convergence rates for zero flux with uh(·, 0) = uI and β1 = 12k(k+1)

.

k N el Order en Order eg Order ‖uh − Ihu‖0 Order

4 3.18e-03 – 2.46e-03 – 4.62e-03 – 5.38e-03 –8 1.94e-04 4.04 1.41e-04 4.12 4.60e-04 3.33 3.44e-04 3.97

2 16 1.20e-05 4.01 8.65e-06 4.03 5.24e-05 3.13 2.15e-05 4.0032 7.49e-07 4.00 5.38e-07 4.00 6.38e-06 3.04 1.35e-06 4.0064 4.68e-08 4.00 3.36e-08 4.00 7.91e-07 3.01 8.41e-08 4.004 6.70e-05 – 2.76e-06 – 3.84e-04 – 9.71e-05 –8 2.36e-06 4.82 7.19e-09 8.58 2.87e-05 3.74 2.08e-06 5.55


4 16 6.90e-10 6.10 5.82e-12 8.03 2.38e-08 5.08 2.47e-10 7.0032 1.06e-11 6.03 2.49e-14 7.87 7.32e-10 5.02 1.93e-12 7.0064 1.66e-13 6.00 8.84e-17 8.13 2.28e-11 5.01 1.50e-14 7.00

TABLE II. Errors and corresponding convergence rates for zero flux with uh(·, 0) = uI and β1 �= 12k(k+1)

.


4 2.76e-03 2.43 2.08e-03 – 1.04e-03 – 3.85e-03 –8 2.59e-04 3.41 1.12e-04 4.22 4.33e-04 1.26 3.60e-04 3.42


4 16 1.33e-07 4.85 1.86e-11 7.86 2.70e-06 3.89 1.70e-07 4.8332 4.26e-09 4.96 7.66e-14 7.93 1.72e-07 3.97 5.49e-09 4.9564 1.34e-10 4.99 3.33e-16 7.84 1.08e-08 4.00 1.73e-10 5.00

We first choose uh(·, 0) = uI (·, 0) with uI (·, 0) defined in (3.40). Note that this special ini-tial discretization satisfies both (3.22) and (3.38). We list in Table I the various errors and thecorresponding convergence rates for k = 2, 3, 4, and T = 1, with the parameter (β0, β1) taken as(10, 1

12 ), (12, 124 ), (16, 1

40 ), respectively. Note that the choice of (β0, β1) assures β1 = 12k(k+1)

for allk = 2, 3, 4. From Table I, we observe a convergence rate k + 2 for el and a rate k + 1 for eg . Theseresults are consistent with the superconvergence results given in Corollary 3.5, which indicatesthat the superconvergence of the derivative approximation at Gauss points and the function valueapproximation at Lobatto points exist. Moreover, the convergence rates provided in (3.27)–(3.28)are optimal. We also observe a superconvergence rate 2k for en when k = 2, 4, which confirms thesuperconvergence results in (3.39). While, it seems that the convergence rate of en can reach 2k+2when k = 3, two-order higher than that in (3.39). As predicted in (3.23), the error ‖uh − Ihu‖0 isconvergent with an order k + 2 for k = 2, 3. While, it seems that the convergence rate of ‖uh−Ihu‖0

is one order better than that in (3.23) when k = 4.To test the influence of β1 on the superconvergence rate, we also consider the case in which

β1 �= 12k(k+1)

. Table II demonstrates the error data and the corresponding convergence rate for

k = 2, 4, with (β0, β1) = (12, 124 ), (10, 1

12 ) respectively. Clearly, we observe a rate k + 1 for el

and ‖uh − Ihu‖0, and a rate k for eg , which means that the superconvergence properties of uh atGauss and Lobatto points, as well as the supercloseness between uh and Ihu, disappear. Therefore,



TABLE III. Errors and corresponding convergence rates for zero flux withuh(·, 0) = Ihu0 andβ1 = 12k(k+1)

.


4 1.22e-02 – 1.75e-03 – 2.34e-02 – 1.63e-02 –8 2.82e-04 5.43 1.39e-04 3.66 9.20e-04 4.67 4.71e-04 5.11



4 16 1.08e-09 6.83 2.31e-14 9.89 3.09e-08 5.67 1.09e-09 7.0632 1.23e-11 6.45 2.56e-17 9.81 7.92e-10 5.28 8.39e-12 7.0264 1.72e-13 6.17 2.94e-20 9.77 2.33e-11 5.09 6.71e-14 6.97

TABLE IV. Errors and corresponding convergence rates for zero flux with uh(·, 0) = Ihu0 and β1 �= 12k(k+1)

.


4 4.14e-03 – 2.36e-03 – 2.19e-03 – 5.07e-03 –8 5.22e-04 2.99 1.39e-04 4.08 1.15e-03 0.93 7.07e-04 2.84


4 16 9.97e-07 4.33 9.53e-11 7.21 2.03e-05 3.36 1.28e-06 4.3132 3.65e-08 4.77 4.46e-13 7.74 1.47e-06 3.78 4.70e-08 4.7664 1.19e-09 4.94 2.21e-15 7.65 9.61e-08 3.94 1.54e-09 4.94

as we indicate in Remark 3.9, the value of β1 affects the superconvergence of uh at Gauss andLobatto points. However, we can still observe the superconvergence phenomenon at nodes, whichindicates that the superconvergence properties of uh at nodes are independent of the choice of β1.

As a comparison group, we also test another way of initial discretization. That is, uh(·, 0) =Ihu0. Note that this choice of initial discretization satisfies the condition (3.30) while doesnot satisfy (3.38). Two cases, i.e. β1 = 1

2k(k+1)and β1 �= 1

2k(k+1), are considered. Listed in

Table III are the corresponding errors and convergence rates in case β1 = 12k(k+1)

. That is, we

choose (β0, β1) = (2, 112 ), (6, 1

24 ), (8, 140 ) for k = 2, 3, 4, respectively. Table IV demonstrates the

corresponding numerical results in case β1 �= 12k(k+1)

for k = 2, 4 with (β0, β1) = (6, 124 ), (2, 1

12 ).Again, we observe similar superconvergence phenomena as in the correction initial disrcretiza-

tion case. To be more precise, when β1 = 12k(k+1)

, the DDG solution is superconvergent with anorder k + 2 to the Gauss–Lobatto projection of the exact solution, as well as the function valueapproximation at Lobatto points; and for the error of derivative approximation at Gauss points, theconvergence rate is k + 1. While the superconvergence phenomena disappears when β1 �= 1

2k(k+1).

It should be pointed out that, we still observe an order 2k for en in both cases β1 = 12k(k+1)

and

β1 �= 12k(k+1)

, although the initial condition (3.38) is not satisfied. In addition, the convergence

rate of en can reach 2k + 2 for k = 3, 4 when β1 = 12k(k+1)

, two-order better than that in (3.39).



TABLE V. Errors and corresponding convergence rates for linear flux with uh(·, 0) = uI and β1 = 12k(k+1)

.


4 6.77e-03 – 5.00e-03 – 2.79e-02 – 8.43e-03 –8 4.08e-04 4.05 3.05e-04 4.04 3.15e-03 3.15 4.61e-04 4.19



4 16 3.10e-08 6.20 1.43e-10 8.06 9.30e-07 5.24 2.29e-08 6.1132 4.67e-10 6.05 5.41e-13 8.04 2.76e-08 5.07 3.52e-10 6.0364 7.21e-12 6.02 2.10e-15 8.01 8.51e-10 5.02 5.47e-12 6.01

TABLE VI. Errors and corresponding convergence rates for linear flux with uh(·, 0) = uI and β1 �= 12k(k+1)

.


4 7.67e-03 – 4.74e-03 – 2.08e-02 – 7.98e-03 –8 6.75e-04 3.51 2.55e-04 4.22 2.41e-03 3.11 6.82e-04 3.55


4 16 3.85e-07 4.75 2.20e-10 7.79 1.09e-05 3.83 3.18e-07 4.5732 1.26e-08 4.93 8.96e-13 7.94 7.00e-07 3.96 1.09e-08 4.8764 4.00e-10 5.00 3.55e-15 7.98 4.41e-08 3.99 3.51e-10 4.96

Example 2. We consider the linear convection-diffusion equation

ut + (αu)x = uxx , (x, t) ∈ (0, 2π) × (0, T ), u(x, 0) = esin(x)/2

with the periodic boundary condition and α = 1 + 12 cos(x − t). The exact solution is

u = esin(x−t)/2.

Listed in Tables V and VI are error data and the corresponding convergence rate for thefinal time T = 1 in cases β1 = 1

2k(k+1)and β1 �= 1

2k(k+1). In Table V, the parameter is cho-

sen as (β0, β1) = (10, 112 ), (12, 1

24 ), (16, 140 ) for k = 2, 3, 4, respectively, and in Table VI,

(β0, β1) = (12, 124 ) for both k = 2, 4. Similar as the zero flux case, we observe a (k + 2) -th order

superconvergence rate for el and ‖uh − Ihu‖0, and a (k + 1) -th order for eg when β1 = 12k(k+1)

.

However, those superconvergence phenomena disappears when β1 �= 12k(k+1)

. Different from the

errors el , eg , we observe a 2k-th superconvergent order for the error en in both β1 = 12k(k+1)

and

β1 �= 12k(k+1)

. As we may recall, all these numerical results are consistent with our theoreticalfindings. Moreover, all the superconvergence rates in (3.27)–(3.28) and (3.39) are optimal, i.e.,the error bounds are sharp.



References

1. B. Cockburn, S. Hou, and C.-W. Shu, The Runge-Kutta local projection discontinuous Galerkin finiteelement method for conservation laws, IV: the multidimensional case, Math Comput 54 (1990), 545–581.

2. B. Cockburn, S. Lin, and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finiteelement method for conservation laws, III: one dimensional systems, J Comput Phys 84 (1989), 90–113.

3. B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite elementmethod for conservation laws, II: general framework, Math Comput, 52 (1989), 411–435.

4. B. Cockburn and C.-W. Shu, The Runge-Kutta local projection P1-discontinuous Galerkin finite elementmethod for scalar conservation laws, RAIRO Modél, Math Anal Numér 25 (1991), 337–361.

5. B. Cockburn and C.-W. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws,V: multidimensional systems, J Comput Phys 141 (1998), 199–224.

6. W. H. Reed and T. R. Hill, Triangular Mesh for Neutron Transport Equation, Los Alamos ScientificLaboratory Report LA-UR-73-479, Los Alamos, NM, 1973.

7. B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J Numer Anal 35 (1998), 2440–2463.

8. D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J NumerAnal 19 (1982), 742–760.

9. G. A. Baker, Finite element methods for elliptic equations using noncomforming elements, Math Comp31 (1977), 45–59.

10. M. F. Wheeler, An elliptic collacation-finite element method with interior penalties, SIAM J NumerAnal 15 (1978), 152–161.

11. C. E. Baumann and J. T. Oden, A discontinuous hp finite element method for convection-diffusionproblems, Comput Methods Appl Mech Eng, 175 (1999), 311–341.

12. J. T. Oden, I. Babuška, and C. E. Baumann, A discontinuous hp finite element method forconvection-diffusion problems, J Comput Phys 146 (1998), 491–519.

13. Y. Cheng and C.-W. Shu, A discontinuous Galerkin finite element method for time dependent partialdifferential equations with higher order derivatives, Math Comput 77 (2007), 699–730.

14. G. Gassner, F. Lörcher, and C. D. Munz, A contribution to the construction of diffusion fluxes for finitevolume and discontinuous Galerkin schemes, J Comput Phys 224 (2007), 1049–1063.

15. B. van Leer and S. Nomura, Discontinuous Galerkin for diffusion, Proceeding of 17th AIAA Compu-tational Fluid Dynamics Conference, Toronto, Ontario Canada, American Institute of Aeronautics andAstronautics, Inc., AIAA-2005-5108, 2005.

16. H. Liu and J. Yan, The direct discontinuous Galerkin (DDG) methods for diffusion problems, SIAM JNumer Anal 47 (2009), 675–698.

17. H. Liu and J. Yan, The direct discontinuous Galerkin (DDG) method for diffusion with interfacecorrections, Commun Comput Phys 8 (2010), 541–564.

18. Y.-Q. Huang, H. Liu, and N.-Y. Yi, Recovery of normal derivatives from the piecewise L2 projection, JComput Phys 231 (2012), 1230–1243.

19. H. Liu, Optimal error estimates of the direct discontinuous Galerkin method for convection-diffusionequations, Math Comput 84 (2015), 2263–2295.

20. D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J NumerAnal 19 (1982), 742C-760C.

21. H. Liu and H. Yu. Maximum-Principle-Satisfying third order discontinuous Galerkin schemes forFokker-Planck equations, SIAM J Sci Comput 36 (2014), A2296C–A2325C.

22. Y.-F. Zhang, X.-X. Zhang, and C.-W. Shu, Maximum-principle-satisfying second order discontinuousGalerkin schemes for convection-diffusion equations on triangular meshes, J Comput Phys 234 (2013),295–316.



23. H. Liu and H. Yu, The entropy satisfying dicontinuous Galerkin method for Fokker-Planck equations,J Sci Comput 62 (2015), 803C–830C.

24. H. Liu and Z.-M. Wang, An entropy satisfying discontinuous Galerkin method for nonlinearFokker-Planck equations, J Sci Comput (2016), 1–24.

25. C. Chen, Structure Theory of Superconvergence of Finite Elements (in Chinese), Hunan Science andTechnology Press, Hunan, China, 2001.

26. R. E. Ewing, R. D. Lazarov, and J. Wang, Superconvergence of the velocity along the Gauss lines inmixed finite element methods, SIAM J Numer Anal 28 (1991), 1015–1029.

27. L. B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics,Vol. 1605, Springer, Berlin, 1995.

28. W. Cao, Z. Zhang, and Q. Zou, Is 2k-conjecture valid for finite volume methods? SIAM J Numer Anal53 (2015), 942–962.

29. C. Chen and S. Hu, The highest order superconvergence for bi-k degree rectangular elements at nodes:a proof of 2k-conjecture, Math Comput 82 (2013), 1337–1355.

30. W. Cao, C.-W. Shu, Y. Yang, and Z. Zhang, Superconvergence of discontinuous Galerkin methods for2-D hyperbolic equations, SIAM J Numer Anal 53 (2015), 1651–1671.

31. W. Cao and Z. Zhang, Superconvergence of local discontinuous Galerkin methods for one-dimensionallinear parabolic equations, Math Comput 85 (2016), 63–84.

32. W. Cao, Z. Zhang, and Q. Zou, Superconvergence of discontinuous Galerkin methods for linearhyperbolic equations, SIAM J Numer Anal 52 (2014), 2555–2573.

33. A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York andLondon, 1966.


superconvergence of the direct discontinuous galerkin method for convection...

Documents