UMER. ANAL Vol. 56, No. 3, pp. 1482{1497szhang/research/p/2018b.pdfSIAM J. NUMER. ANAL. c 2018...

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SIAM J. NUMER. ANAL. c© 2018 Society for Industrial and Applied MathematicsVol. 56, No. 3, pp. 1482–1497

A WEAK GALERKIN FINITE ELEMENT METHOD FORSINGULARLY PERTURBED

CONVECTION-DIFFUSION–REACTION PROBLEMS∗

RUNCHANG LIN† , XIU YE‡ , SHANGYOU ZHANG§ , AND PENG ZHU¶

Abstract. In this article, a new weak Galerkin finite element method is introduced to solveconvection-diffusion–reaction equations in the convection dominated regime. Our method is highlyflexible by allowing the use of discontinuous approximating functions on polytopal mesh withoutimposing extra conditions on the convection coefficient. An error estimate is devised in a suitablenorm. Numerical examples are provided to confirm theoretical findings and efficiency of the method.

Key words. weak Galerkin, finite element method, discrete gradient, singular perturbation,convection-diffusion reaction, polyhedral mesh

AMS subject classifications. Primary, 65N15, 65N30; Secondary, 35J50

DOI. 10.1137/17M1152528

1. Introduction. In this work, we consider a finite element method (FEM) forthe following convection-diffusion–reaction problems exhibiting layer behavior

−ε∆u+∇ · (bu) + cu = f in Ω ⊂ Rd,(1.1)

u = 0 on ∂Ω,(1.2)

where ε > 0 is a parameter, d = 2 or 3, and Ω is a polygonal (when d = 2) or polyhe-dral (when d = 3) domain with boundary ∂Ω. For well-posedness of the differentialequation, we assume that b, c, and f are sufficiently smooth, b ∈ [W 1,∞(Ω)]d, andc+ 1

2∇ · b ≥ c0 > 0 for some constant c0 (cf., e.g., [32]).It is well known that the linear elliptic equation (1.1)–(1.2), though it looks sim-

ple, is very difficult to solve when it is singularly perturbed (i.e., when the singularperturbation parameter ε 1). Due to the very small diffusion coefficient, the solu-tion of the boundary value problem typically possesses layers, which are thin regionswhere the solution and/or its derivatives change rapidly. Standard numerical methodsfail to provide accurate approximations in this case unless the computational mesh isof the magnitude of the layers.

Numerical stabilization techniques have been developed to resolve the difficulty,which can be roughly classified into fitted mesh methods and fitted operator methods.Efforts toward the optimization of numerical meshes can be traced back to the works of

∗Received by the editors October 17, 2017; accepted for publication (in revised form) January 29,2018; published electronically May 24, 2018.

http://www.siam.org/journals/sinum/56-3/M115252.htmlFunding: The work of the first author was partially supported by a University Research Grant

of Texas A&M International University. The work of the second author was supported in part by theNational Science Foundation under grant DMS-1620016. The work of the fourth author was partiallysupported by Zhejiang Provincial Natural Science Foundation of China (grant LY15A010018).†Department of Mathematics and Physics, Texas A&M International University, Laredo, TX

78041 ([email protected]).‡Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204

([email protected]).§Department of Mathematical Sciences, University of Delaware, Newark, DE 19716 (szhang@udel.

edu).¶School of Mathematics, Physics and Information, Jiaxing University, Jiaxing, Zhejiang 314001,

China ([email protected]).

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mailto:[email protected]






A WG FEM FOR CONVECTION-DIFFUSION–REACTION PROBLEMS 1483

Bakhvalov [5], whose meshes are obtained from projections of an equidistant partitionof layer functions. An easier yet efficient idea of piecewise-equidistant meshes proposedby Shishkin [34] has attracted tremendous attention and set a trend of studies ofproblems with singular perturbation, which is still in vogue. In addition, adaptivemethods have been gaining popularity for more than four decades [4] (cf. also [11]),which have addressed a variety of difficulties including layers [30]. For details of fittedmesh methods on singularly perturbed problems, we refer to the books [21, 22, 32, 35]and the references therein.

When solving the problems of complicated domains or layer structures, fitted op-erator methods, dominated by upwind-type schemes, are usually applied. The mecha-nism of upwinding is to add artificial diffusion/viscosity to balance the convection andto stabilize a standard discretization method. Upwind schemes were first proposed asfinite difference methods, which were later extended to FEMs. Among variations ofapproaches, the streamline upwind/Petrov–Galerkin (SUPG) method introduced byHughes and Brooks is one of the most successful methods [18]; see also [7]. The SUPGmethod improves stability of the standard Galerkin method by adding residuals withdiscontinuous weights. A drawback of upwind methods is that too much artificialdiffusion will smear layers, which is a particular concern for the problems in multipledimensions. For more discussions about upwind methods, please refer to [32] and itsreferences.

In the last two decades, the flexibility and special capabilities of utilizing dis-continuous discrete trial and test spaces have prompted vigorous developments of avariety of methods, including discontinuous Galerkin (DG) methods, weak Galerkin(WG) methods, etc. The DG method, also known as the boundary penalty methodand interior penalty method in different contexts, originated in the early 1970s; see,e.g., [3, 14, 28, 29, 39]. It has been a major technique for solving conservation laws,which later became a very popular approach for elliptic problems [1]. When appliedto convection-diffusion problems, the DG approach includes a natural upwinding thatis equivalent to some stabilization [20, 32]. Literature on DG methods for convection-diffusion problems includes [2, 6, 8, 15, 17, 19]. For detailed discussions on DG meth-ods, we refer to [12, 16, 31]. The WG method is a recently developed novel frameworkfor solving partial differential equations [36]. With new concepts referred to as weakfunction and weak gradient, the numerical approach allows the use of totally discon-tinuous functions and has a simple formulation which is parameter independent. TheWG method has been rapidly developed and applied to different types of problems,including second order elliptic problems, biharmonic problems, and Stokes flow, etc.;see, e.g., [23, 24, 25, 26, 27, 36, 38]. WG methods allow adoption of general polytopalmeshes, which provide enormous flexibility in scheme design and implementation. Re-cently, a WG method was studied in [9] for problem (1.1)–(1.2) with a set of strictassumptions on data of the differential equation.

The goal of this article is to develop a new simple WG method for the convection-diffusion–reaction problem (1.1)–(1.2) with singular perturbation. The proposedmethod is also a fitted operator-type method, which yields stable numerical solutionsover uniform meshes. Our WG method, like the DG method, involves upwinding byusing discontinuous approximations across elements but not artificial residuals, whichhence requires more degrees of freedom than many other upwind-type schemes, suchas the SUPG method.

Compared to many existing methods, our method has some unique features: (1)A different stabilization technique is used to avoid smearing the layers by over sta-bilization. (2) A set of strict assumptions for the convection coefficient, in [9] and

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1484 RUNCHANG LIN, XIU YE, SHANGYOU ZHANG, AND PENG ZHU

other papers, is removed. (3) Error analysis has been significantly simplified. Ourerror analysis is clean and short. (4) The discontinuity between elements makes itpossible to use polytopal meshes (see section 2), which allow remarkable conveniencein adaptive mesh generation and implementation; cf., e.g., [10, 13, 24, 25]. Whenshape-regular meshes are used, a priori error estimates of O(hk+1/2) for the kth or-der element can be obtained. Extensive numerical experiments have been conducted.Numerical results show that the proposed WG method is efficient for different typesof singularly perturbed convection-diffusion–reaction problems.

Throughout this article, we use C for generic constants independent of ε, meshsize, and the solution to (1.1)–(1.2), which may not necessarily be the same at eachoccurrence.

The rest of this article is organized as follows. In section 2, a new WG FEM isproposed. Analysis of the new method is found in section 3. Numerical experimentsare presented in section 4 to support the theoretical results.

2. WG finite element schemes. For any given polyhedron D ⊆ Ω, we usethe standard definition of Sobolev spaces Hs(D) with s ≥ 0. The associated innerproduct, norm, and seminorms in Hs(D) are denoted by (·, ·)s,D, ‖ · ‖s,D, and | ·|s,D, respectively. When s = 0, H0(D) coincides with the space of square integrablefunctions L2(D). In this case, the subscript s is suppressed from the notation of norm,seminorm, and inner products. Furthermore, the subscript D is also suppressed whenD = Ω.

Let Th be a partition of the domain Ω consisting of elements which are closed andsimply connected polygons in two dimensions or polyhedra in three dimensions; seeFigure 1. Let Eh be the set of all edges or flat faces in Th, and E0

h = Eh\∂Ω be theset of all interior edges or flat faces. Denote by hT the diameter for every elementT ∈ Th and h = maxT∈Th hT the mesh size for Th. We need some shape regularityassumptions for the partition Th described as below (cf. [37]).

A1: Assume that there exist two positive constants %v and %e such that for everyelement T ∈ Th we have

(2.1) %vhdT ≤ |T |, %eh

d−1e ≤ |e|

for all edges or flat faces of T .

xe

Ae

A

B

C

D

EF

n

Fig. 1. Depiction of a shape-regular polygonal element ABCDEFA.

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A2: Assume that there exists a positive constant κ such that for every elementT ∈ Th we have

(2.2) κhT ≤ he

for all edges or flat faces e of T .A3: Assume that the mesh edges or faces are flat. We further assume that for every

T ∈ Th, and for every edge/face e ∈ ∂T , there exists a pyramid P (e, T,Ae)contained in T such that its base is identical with e, its apex is Ae ∈ T , andits height is proportional to hT with a proportionality constant σe boundedaway from a fixed positive number σ∗ from below. In other words, the heightof the pyramid is given by σehT such that σe ≥ σ∗ > 0. The pyramid is alsoassumed to stand up above the base e in the sense that the angle betweenthe vector xe − Ae, for any xe ∈ e, and the outward normal direction of e isstrictly acute by falling into an interval [0, θ0] with θ0 <

π2 .

A4: Assume that each T ∈ Th has a circumscribed simplex S(T ) that is shaperegular and has a diameter hS(T ) proportional to the diameter of T , i.e.,hS(T ) ≤ γ∗hT with a constant γ∗ independent of T . Furthermore, assumethat each circumscribed simplex S(T ) intersects with only a fixed and smallnumber of such simplices for all other elements T ∈ Th.

For a given integer k ≥ 1, let Vh be the WG finite element space associated withTh defined as follows:

(2.3) Vh = v = v0, vb : v0|T ∈ Pk(T ), vb|e ∈ Pk(e), e ⊂ ∂T , T ∈ Th

and

(2.4) V 0h = v : v ∈ Vh, vb = 0 on ∂Ω,

where Pk is the space of polynomials of total degree k or less. We would like toemphasize that any function v = v0, vb ∈ Vh has a single value vb on each edgee ∈ Eh, which is not necessarily the trace of v0 on e. The discontinuity betweenelements allows us to apply Pk spaces on polytopal elements, which brings convenienceto many aspects in implementation.

For any v = v0, vb, a weak gradient ∇wv ∈ [Pk−1(T )]d is defined on T as theunique polynomial satisfying

(2.5) (∇wv, τ)T = −(v0, ∇ · τ)T + 〈vb, τ · n〉∂T ∀τ ∈ [Pk−1(T )]d,

where n is the unit outward normal vector to ∂T , and (·, ·)T and 〈·, ·〉∂T are standardL2 inner products on T and ∂T , respectively. For any v = v0, vb, we define a weakdivergence ∇w · (bv) ∈ Pk(T ) related to b on T as the unique polynomial satisfying

(2.6) (∇w · (bv), w)T = −(bv0, ∇w)T + 〈b · nvb, w〉∂T ∀w ∈ Pk(T ).

We introduce four global projections Q0, Qb, Qh, and Qh. They are element-wise defined L2 projections detailed as follows. For each element T ∈ Th, Q0 :L2(T ) → Pk(T ) and Qb : L2(e) → Pk(e) are the L2 projections onto the associatedlocal polynomial spaces, Qh: [L2(T )]d → [Pk−1(T )]d is the L2 projection onto the localweak gradient space. Finally, we define a projection operator Qhu = Q0u,Qbu ∈ Vhfor the true solution u.

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For simplicity, we adopt the following notations,

(v, w)Th =∑T∈Th

(v, w)T =∑T∈Th

∫T

vwdx,

〈v, w〉∂Th =∑T∈Th

〈v, w〉∂T =∑T∈Th

∫∂T

vwds.

For v = v0, vb and w = w0, wb in Vh, we define a bilinear form as

(2.7) a(v, w) = ε(∇wv,∇ww)Th + (∇w · (bv), w0)Th + (cv0, w0) + sc(v, w) + sd(v, w),

where

sc(v, w) =∑T∈Th

〈b · n(v0 − vb), w0 − wb〉∂+T ,

sd(v, w) =∑T∈Th

εh−1T 〈v0 − vb, w0 − wb〉∂T ,

and

∂+T = x ∈ ∂T : b(x) · n(x) ≥ 0.

Then a WG FEM is proposed in the following.

Algorithm 1 WG method.

A WG approximation for (1.1)–(1.2) is to seek uh = u0, ub ∈ V 0h satisfying the

following equation:

(2.8) a(uh, v) = (f, v0) ∀ v = v0, vb ∈ V 0h .

Accordingly, we define an energy norm ||| · ||| in V 0h : for any v ∈ V 0

h ,

(2.9) |||v|||2 = ε∑T∈Th

‖∇wv‖2T +∑T∈Th

∥∥∥|b · n|1/2(v0 − vb)∥∥∥2

∂T+ ‖v0‖2 + sd(v, v).

Remark 1. Compared to the WG scheme proposed in [9], our stabilizer for con-vection term sc(·, ·) is simple and has upwinding flavor. Moreover, unlike in [2, 9], ourWG method does not make extra assumptions on the convection coefficient b.

3. Analysis. In this section, we will establish the well-posedness of the WGmethod (2.8) and obtain an error estimate for the WG finite element approximationuh.

3.1. Well-posedness of the WG method and error equation. Verificationof continuity of the bilinear form (2.7) is straightforward. We next show the coercivityof the bilinear form. The following lemma is useful.

Lemma 3.1. For v, w ∈ V 0h ,

(∇w · (bv), w0)Th = (∇ · bv0, w0)Th − (v0,∇w · (bw))Th− 〈b · n(v0 − vb), w0 − wb〉∂Th .(3.1)

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Proof. It follows from the definition of the weak divergence (2.6) that

(∇w · (bv), w0)Th = −(bv0,∇w0)Th + 〈b · nvb, w0〉∂Th= (∇ · bv0, w0)Th + (bw0,∇v0)Th − 〈b · n(v0 − vb), w0〉∂Th= (∇ · bv0, w0)Th − (∇w · (bw), v0)Th + 〈b · nwb, v0〉∂Th− 〈b · n(v0 − vb), w0〉∂Th

= (∇ · bv0, w0)Th − (∇w · (bw), v0)Th− 〈b · n(v0 − vb), w0 − wb〉∂Th ,

which implies (3.1). Here we use the facts that v, w ∈ V 0h and 〈b · nvb, wb〉∂Th = 0 to

obtain the last equation above.

Lemma 3.2. For v ∈ V 0h , then

(3.2) C|||v|||2 ≤ a(v, v).

Therefore, the WG formulation (2.8) has a unique solution.

Proof. It follows from (3.1) that for any v ∈ V 0h ,

(3.3) (∇w · (bv), v0)Th =1

2(∇ · bv0, v0)Th −

1

2〈b · n(v0 − vb), v0 − vb〉∂Th .

Using (3.3), we have

a(v, v) = ε(∇wv,∇wv)Th +

((c+

1

2∇ · b

)v0, v0

)Th

− 1

2〈b · n(v0 − vb), v0 − vb〉∂Th + sc(v, v) + sd(v, v)

≥ ε(∇wv,∇ww)Th + c0‖v0‖2 +1

2

∑T∈Th

∥∥∥|b · n|1/2(v0 − vb)∥∥∥2

∂T+ sd(v, v)

≥ C|||v|||2,

which completes the proof.

We next derive an equation that the error satisfies, which will be used in erroranalysis below.

Lemma 3.3. Let u be the solution of the problem (1.1)–(1.2). Then for v ∈ V 0h ,

(∇ · (bu), v0) = (∇w · (bQhu), v0)Th − `1(u, v) + `2(u, v),(3.4)

where

`1(u, v) = (u−Q0u,b · ∇v0)Th ,

`2(u, v) = 〈u−Qbu,b · n(v0 − vb)〉∂Th .

Proof. It follows from the definition of the weak divergence (2.6) that

(∇ · (bu), v0)Th = −(bu,∇v0)Th + 〈b · nu, v0〉∂Th= −(bQ0u,∇v0)Th − `1(u, v) + 〈b · nQbu, v0〉∂Th− 〈b · nQbu, v0〉∂Th + 〈b · nu, v0 − vb〉∂Th

= (∇w · (bQhu), v0)Th − `1(u, v) + `2(u, v),

which implies (3.4).

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−ε(∆u, v0) = ε(∇wQhu,∇wv)Th − à(u, v),(3.5)

where

à(u, v) = ε〈(∇u−Qh∇u) · n, v0 − vb〉∂Th .

Proof. We refer the proof of the lemma to [25].

Let `b(u, v) = `1(u, v) − `2(u, v) and `c(u, v) = −(cu − cQ0u, v0). We have thefollowing error equation.

Lemma 3.5. Let eh = Qhu− uh ∈ V 0h . Then, for any v ∈ V 0

h we have

(3.6) a(eh, v) = à(u, v) + `b(u, v) + `c(u, v) + sc(Qhu, v) + sd(Qhu, v).

Proof. Testing (1.1) by v = v0, vb ∈ V 0h , we arrive at

(3.7) − ε(∆u, v0) + (∇ · (bu), v0) + (cu, v0) = (f, v0).

Using (3.4) and (3.5), (3.7) becomes

ε(∇wQhu,∇wv)Th + (∇w · (bQhu), v0)Th + (cQ0u, v0)

= (f, v0) + à(u, v) + `b(u, v) + `c(u, v).

Adding sc(Qhu, v) and sd(Qhu, v) to both sides of the above equation gives

(3.8) a(Qhu, v) = (f, v0) + à(u, v) + `b(u, v) + `c(u, v) + sc(Qhu, v) + sd(Qhu, v).

Subtracting (2.8) from (3.8) yields the error equation (3.6). This completes the proofof the lemma.

3.2. Error estimates. We will derive error estimates in this section. For anyfunction ϕ ∈ H1(T ), the following trace inequality holds true:

(3.9) ‖ϕ‖2e ≤ C(h−1T ‖ϕ‖

2T + hT ‖∇ϕ‖2T

).

Lemma 3.6. Let u be the solution of the problem (1.1)–(1.2) and Th be a finiteelement partition of Ω satisfying the shape regularity assumptions A1–A4. Then, theL2 projections Q0 and Qh satisfy∑

T∈Th

(‖u−Q0u‖2T + h2

T ‖∇(u−Q0u)‖2T)≤ Ch2(s+1)‖u‖2s+1, 0 ≤ s ≤ k,

∑T∈Th

(‖∇u−Qh∇u‖2T + h2

T |∇u−Qh∇u|21,T)≤ Ch2s‖u‖2s+1, 0 ≤ s ≤ k.

Proof. In order to avoid repetition, we refer the readers to the proof of [37,Lemma 4.1].


|à(u, v)| ≤ Cε 12hk|u|k+1|||v|||,(3.10)

|`b(u, v)| ≤ Chk+ 12 |u|k+1|||v|||,(3.11)

|`c(u, v)| ≤ Chk+1|u|k+1|||v|||,(3.12)

|sc(Qhu, v)| ≤ Chk+ 12 |u|k+1|||v|||,(3.13)

|sd(Qhu, v)| ≤ Cε 12hk|u|k+1|||v|||.(3.14)

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Proof. It follows from the Cauchy–Schwarz inequality, the trace inequality (3.9),and Lemma 3.6 that

|`a(u, v)| ≤∑T∈Th

∣∣〈ε(∇u−Qh∇u) · n, v0 − vb〉∂T∣∣

≤ C∑T∈Th

ε‖∇u−Qh∇u‖∂T ‖v0 − vb‖∂T

≤ C

(∑T∈Th

hT ε‖∇u−Qh∇u‖2∂T

) 12(∑T∈Th

h−1T ε‖v0 − vb‖2∂T

) 12

≤ Cεhk|u|k+1

(∑T∈Th

εh−1T ‖v0 − vb‖2∂T

) 12

≤ Cεhk|u|k+1|||v|||.

To prove (3.11), we need to bound `1(u, v) and `2(u, v) by the definition of `b(u, v). Letb be a piecewise constant function whose value is the average of b over the element.It follows from Lemma 3.6 and the inverse inequality,

|`1(u, v)| = |(u−Q0u,b · ∇v0)Th | =∣∣(u−Q0u, (b− b) · ∇v0)Th

∣∣≤ Chk+1|u|k+1‖v0‖.

On the other hand, it follows from the definition of Qb, Lemma 3.6, and the Cauchy–Schwarz inequality,

|`2(u, v)| =∣∣〈u−Qbu,b · n(v0 − vb)〉∂Th

∣∣≤ Chk+ 1

2 |u|k+1|||v|||,

where we use the fact ‖u−Qbu‖e ≤ ‖u−Q0u‖e.Combining the two estimates above, we have proved (3.11). Similarly, we can

obtain (3.12).It follows from (3.9), the Cauchy–Schwarz inequality, and Lemma 3.6,

|sc(Qhu, v)|

≤∑T∈Th

∣∣∣〈b · n(Q0u−Qbu), v0 − vb〉∂+T∣∣∣

≤

(∑T∈Th

∥∥∥|b · n|1/2(Q0u− u+ u−Qbu)∥∥∥2

∂T

)1/2(∑T∈Th

∥∥∥|b · n|1/2(v0 − vb)∥∥∥2

∂T

)1/2

≤ C

(∑T∈Th

∥∥∥|b · n|1/2(Q0u− u)∥∥∥2

∂T

)1/2(∑T∈Th

∥∥∥|b · n|1/2(v0 − vb)∥∥∥2

∂T

)1/2

≤ Chk+ 12 |u|k+1|||v|||.

Similarly, we can obtain (3.14) and complete the proof of the lemma.

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Theorem 3.8. Let uh ∈ V 0h be the WG finite element solution of the problem

(1.1)–(1.2) arising from (2.8). Then there exists a constant C such that

(3.15) |||Qhu− uh||| ≤ C(ε1/2 + h1/2

)hk|u|k+1.

Proof. It follows from (3.2) that

|||Qhu− uh|||2 ≤ Ca(eh, eh).

Letting v = eh in (3.6) gives

a(eh, eh) = `a(u, eh) + `b(u, eh) + `c(u, eh) + sc(Qhu, eh) + sd(Qhu, eh).

Combining the two equations above with Lemma 3.7, we have

|||Qhu− uh||| ≤ C(ε1/2 + h1/2

)hk|u|k+1.

We have proved the theorem.

Next, we discuss the relation between the WG norm ||| · ||| defined in (2.9),

|||v|||2 = ε∑T∈Th

‖∇wv‖2T + ‖v0‖2 +∑T∈Th

∥∥∥|b · n|1/2(v0 − vb)∥∥∥2

∂T+ sd(v, v),

and the SUPG norm |||·|||SUPG defined as follows for a weak function v = v0, vb ∈ Vh,

(3.16) |||v|||2SUPG = ε∑T∈Th

‖∇v0‖2T + ‖v0‖2 +∑T∈Th

δT ‖b · ∇v0‖2T .

The following lemma is helpful in understanding the relation between the twonorms defined above.

Lemma 3.9. There exists a positive constant C such that for any v = v0, vb ∈Vh, we have

(3.17)∑T∈Th

‖∇v0‖2T ≤ C

(∑T∈Th

‖∇wv‖2T +∑T∈Th

h−1T 〈v0 − vb, v0 − vb〉∂T

).

Proof. For any v = v0, vb ∈ Vh, it follows from the definition of the weakgradient (2.5) and integration by parts that for any q ∈ [Pk−1(T )]d,

(∇wv, q)T = −(v0,∇ · q)T + 〈vb, q · n〉∂T = (∇v0, q)T + 〈vb − v0, q · n〉∂T .

By letting q = ∇v0 in the above identity we arrive at

(∇wv,∇v0)T = (∇v0,∇v0)T + 〈vb − v0,∇v0 · n〉∂T .

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Thus, from the Cauchy–Schwarz, trace, and inverse inequalities, we have

‖∇v0‖2T ≤ ‖∇wv‖T ‖∇v0‖T + Ch−1/2T ‖v0 − vb‖∂T ‖∇v0‖T .

This leads to

‖∇v0‖T ≤ C(‖∇wv‖2T + Ch−1

T ‖v0 − vb‖2∂T) 1

2 ,

which completes the proof of the lemma.

Lemma 3.9 implies that for v = v0, vb ∈ Vh,

(3.18) ε∑T∈Th

‖∇v0‖2T ≤ C

(ε∑T∈Th

‖∇wv‖2T + sd(v, v)

).

Remark 2. It follows from (3.18) that the two norms |||·||| and |||·|||SUPG are similar,where the only difference is that the term

∑T∈Th δT ‖b ·∇v0‖2T in ||| · |||SUPG is replaced

by∑T∈Th ‖|b ·n|

1/2(v0−vb)‖2∂T in ||| · |||. This difference reflects different “upwinding”strategies of the SUPG method and the WG method. The SUPG method adds aresidual term to the continuous FEM to provide additional control on the convectivederivative in the streamline direction. On the other hand, the term

∑T∈Th ‖|b ·

n|1/2(v0 − vb)‖2∂T in ||| · ||| represents the upwinding strategy of using discontinuousapproximation.

4. Numerical example. In this section, we test numerically the performanceof the proposed WG FEM for solving boundary value problems (1.1)–(1.2). Examplesof different types have been tested to confirm the efficiency of the method. Numericalerrors will be measured in the L2 norm ‖ · ‖ and/or the energy norm ||| · |||.

4.1. Example 1: Smooth solution. This example is adopted from [2]. LetΩ = (0, 1)2, b = (1, 1)T , and c = 0. The source term f is chosen so that the exactsolution is

u(x, y) = sin(2πx) sin(2πy).

We computed approximate solutions to the boundary value problem for ε = 10−3

and 10−9 to confirm that the WG scheme is valid. Triangular meshes are used.Convergence plots for numerical errors for linear, quadratic, and cubic elements havebeen collected in Figure 2. It is observed that an optimal L2 estimate of O(hk+1) isobtained. On the other hand, |||Qhu − uh||| converges at O(hk+1/2), which matchesthe estimation in Theorem 3.8.

4.2. Example 2: Boundary layer. In this example, we examine the perfor-mance of the proposed WG method in the occurrence of the boundary layer. LetΩ = (0, 1)2,b = (1, 1)T , and c = 0 in (1.1). We choose f and an appropriate Dirichletboundary condition so that the exact solution is

(4.1) u(x, y) = sinπx

2+ sin

πy

2+(e−1/ε − e(x−1)(y−1)/ε

)/(

1− e−1/ε).

P2 finite elements are used on rectangular uniform meshes. Numerical errors measuredin different norms and their convergence rates are provided in Tables 1 and 2.

In particular, as shown in Table 1, the optimal convergence rate of O(hk+1/2) inTheorem 3.8 is confirmed. For comparison, we measure also the errors of the WG

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(a) ‖Q0u− u0‖, ε = 10−3 (b) ‖Q0u− u0‖, ε = 10−9

(c) |||Qhu− uh|||, ε = 10−3 (d) |||Qhu− uh|||, ε = 10−9

Fig. 2. Example 1. Convergence rates of ‖Q0u − u0‖ and of |||Qhu − uh|||of P1, P2, and P3

elements.

Table 1Example 2. The numerical errors and their orders of convergence with P2 elements when

ε = 10−9.

|||Qhu− uh||| hn |||Qhu− uh|||SUPG hn

1 0.0692366 0.12143352 0.0177851 2.26 0.0269782 2.503 0.0034200 2.57 0.0049298 2.654 0.0006320 2.54 0.0009027 2.555 0.0001142 2.52 0.0001631 2.526 0.0000204 2.51 0.0000292 2.51

finite element solutions in the standard SUPG energy norm [18] defined by (3.16) (seealso [32]). In Table 1, we present the numerical results with δT = hT in (3.16). It isobserved that the convergence order of Qhu − uh in the SUPG norm is the same asthat measured in the WG energy norm.

We shall report that, as in other upwind schemes (see, e.g., [32]), our methodhas disappointing performance inside the layer; cf. also Remark 1. In Table 2, itis observed that the numerical behavior of the method is poor in the intermediateregimes (e.g., when ε = 10−3), which, however, is very good in the strongly advection-dominated cases (e.g., when ε = 10−9); see also Table 1. Similar phenomena in the

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Table 2Example 2. The numerical errors and their orders of convergence with P2 elements.

ε = 10−3 ε = 10−9

‖Q0u− u0‖0 hn ‖Q0u− u0‖0 hn |Q0u− u0|1,h hn

1 0.0040535 0.08056691 0.35863322 0.0028010 0.60 0.00516027 4.57 0.0460337 3.413 0.0111441 0.00 0.00078110 2.95 0.0090941 2.534 0.0115248 0.00 0.00005975 3.87 0.0019822 2.295 0.0083321 0.48 0.00000750 3.06 0.0004856 2.076 0.0046727 0.84 0.00000095 3.00 0.0001212 2.02

(a) Numerical solution (b) Exact solution

Fig. 3. Example 2. The numerical and exact solutions using P2 elements on a mesh of 1,024elements, ε = 10−3.

(a) Numerical solution (b) Exact solution

Fig. 4. Example 2. The numerical and exact solutions using P2 elements on a mesh of 1,024elements, ε = 10−9.

DG method have been reported in the literature (see, e.g., [2, 19]). On the otherhand, when ε = 10−9, optimal convergence of Q0u− u0 in both the L2 norm and theH1 seminorm is obtained.

The plots of uh for ε = 10−3 and ε = 10−9 over a mesh of 1,024 uniform elementsare shown in Figures 3 and 4, respectively. In Figures 3 and 4, u0 is plotted for thenumerical solutions uh = u0, ub. For the exact solution u, Q0u ofQhu = Q0u,Qbuis plotted and the average of Q0u and Qbu is plotted on ∂Ω.

4.3. Example 3: Interior layer—continuous boundary conditions. Inthis example, we examine the performance of the proposed WG method in the occur-rence of an interior layer. Let Ω = (0, 1)2, b = (1, 0)T , and c = 1. The source term fis given such that the exact solution is

u(x, y) = 0.5x(1− x)y(1− y)

(1− tanh

α− xγ

).

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(a) Exact solution (b) Numerical solution over a 32 × 32 × 2uniform triangle mesh

Fig. 5. Example 3. Exact solution and WG solution of P1 element with ε = 10−9.

10−2

10−1

100

10−6

10−5

10−4

10−3

10−2

10−1

100

ε = 10−9

|||Qhu−u

h|||

O(h3/2)||Q

0u−u

0||

O(h2)

Fig. 6. Example 3. Convergence rates of |||Qhu − uh||| and ‖Q0u − u0‖ of P1 element withε = 10−9.

Here the parameters α and γ control the location and thickness of the interior layer.In Figures 5 and 6, numerical results of the convection-diffusion–reaction problemwith ε = 10−9, α = 0.5, and γ = 0.05 are provided. When a P1 finite element spaceis used, the numerical errors |||Qhu − uh||| and ‖Q0u − u0‖ converge in O(h3/2) andO(h2), respectively. The interior layer can be accurately captured.

4.4. Example 4: Internal layer—discontinuous boundary conditions.Let Ω = (0, 1)2, b = (1, 1)T , and c = 0. We consider the following problem

− ε∆u+∇ · (bu) + cu = 0 in Ω,(4.2)

u = g =

1 on 0 × (1/4, 1),

0 elsewhere on ∂Ω.(4.3)

Due to the discontinuities in boundary conditions, the solution of this problem is notin H1(Ω). Numerical oscillations occur near the interior layer caused by the joints ofthe conflicting Dirichlet boundary conditions. This phenomenon has been reportedin the literature for DG methods; see, e.g., [2, 19].

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(a) uh by P2 element (b) uh by P4 element

Fig. 7. Example 4. The solution uh by (2.8) on a mesh of 1,024 uniform squares.

(a) uh by P2 element (b) uh by P4 element

Fig. 8. Example 4. The solution uh by (4.4) on a mesh of 1,024 uniform squares.

We use Pk elements over rectangular meshes. The discrete solutions to problem(4.2)–(4.3) for k = 2 and k = 4 on the mesh of 1,024 uniform squares are plotted inFigure 7, in which oscillations are observed near the interior layer. We believe that abetter numerical approximation can be obtained if the Dirichlet boundary conditionsare enforced weakly. That is, the FEM (2.8) is modified as finding uh = (u0, ub) ∈ Vhsuch that

a(uh, v) +ε

h〈ub, vb〉∂Ω − ε〈∇wuh · n, vb〉∂Ω = (f, v0) +

ε

h〈g, vb〉∂Ω ∀v ∈ Vh.(4.4)

The solutions to (4.4) using P2 and P4 elements are shown in Figure 8. Improvedresults are observed. This phenomenon has been discussed in the literature; see,e.g., [33]. For both (2.8) and (4.4), the solutions do not oscillate out of the internallayer.

4.5. Example 5: Rotational flow. We solve the following boundary valueproblem

−ε∆u+∇ · (bu) + cu = 0 in Ω = (0, 1)2 \

1

2

×(

0,1

2

),

u = x(1− x)y(1− y)

(y − 1

2

)2

on ∂Ω,

(4.5)

where ε = 10−9, b = ( 12−y, x−

12 ), and c = 10−3. Pk elements are used on rectangular

meshes. The WG solutions for k = 2 and k = 4 over a mesh of 256 elements are plottedin Figure 9. The numerical results demonstrate that the WG discretization is stable.

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(a) k = 2 (b) k = 4

Fig. 9. Example 5. The numerical solution uh on a mesh of 256 uniform squares.

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