A Weak Galerkin Finite Element Method for Singularly Perturbed Convection-Diffusion–Reaction Problems∗

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

A WEAK GALERKIN FINITE ELEMENT METHOD FOR SINGULARLY 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 solve convection-diffusion–reaction equations in the convection dominated regime. Our method is highly flexible by allowing the use of discontinuous approximating functions on polytopal mesh without imposing extra conditions on the convection coefficient. An error estimate is devised in a suitable norm. 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) for

the following convection-diﬀusion–reaction problems exhibiting layer behavior d (1.1) −∆u + ∇ · (bu) + cu = f in Ω ⊂ R , (1.2) u = 0 on ∂Ω,

where > 0 is a parameter, d = 2 or 3, and Ω is a polygonal (when d = 2) or polyhedral (when d = 3) domain with boundary ∂Ω. For well-posedness of the diﬀerential

equation, we assume that b, c, and f are sufficiently smooth, b ∈ [W 1,∞(Ω)]d, and 1 c + 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 simple, is very difficult to solve when it is singularly perturbed (i.e., when the singular perturbation parameter 1). Due to the very small diffusion coefficient, the solu-

tion of the boundary value problem typically possesses layers, which are thin regions where the solution and/or its derivatives change rapidly. Standard numerical methods fail to provide accurate approximations in this case unless the computational mesh is of 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.html Funding: 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 the National Science Foundation under grant DMS-1620016. The work of the fourth author was partially supported 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.

Downloaded 08/29/18 to 128.175.16.112. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php edu). ¶School of Mathematics, Physics and Information, Jiaxing University, Jiaxing, Zhejiang 314001, China ([email protected]).

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Bakhvalov [5], whose meshes are obtained from projections of an equidistant partition of layer functions. An easier yet eﬃcient idea of piecewise-equidistant meshes proposed by Shishkin [34] has attracted tremendous attention and set a trend of studies of problems with singular perturbation, which is still in vogue. In addition, adaptive methods 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 fitted mesh 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 operator methods, dominated by upwind-type schemes, are usually applied. The mecha-

nism of upwinding is to add artificial diffusion/viscosity to balance the convection and to stabilize a standard discretization method. Upwind schemes were first proposed as finite difference methods, which were later extended to FEMs. Among variations of approaches, the streamline upwind/Petrov–Galerkin (SUPG) method introduced by Hughes and Brooks is one of the most successful methods [18]; see also [7]. The SUPG

method improves stability of the standard Galerkin method by adding residuals with discontinuous weights. A drawback of upwind methods is that too much artiﬁcial diﬀusion will smear layers, which is a particular concern for the problems in multiple dimensions. For more discussions about upwind methods, please refer to [32] and its references.

In the last two decades, the ﬂexibility and special capabilities of utilizing discontinuous discrete trial and test spaces have prompted vigorous developments of a variety of methods, including discontinuous Galerkin (DG) methods, weak Galerkin (WG) methods, etc. The DG method, also known as the boundary penalty method and 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 applied to convection-diﬀusion problems, the DG approach includes a natural upwinding that is 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 framework for solving partial differential equations [36]. With new concepts referred to as weak function and weak gradient, the numerical approach allows the use of totally discontinuous functions and has a simple formulation which is parameter independent. The WG 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 polytopal meshes, 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 strict assumptions 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 proposed method is also a fitted operator-type method, which yields stable numerical solutions over uniform meshes. Our WG method, like the DGmethod, involves upwinding by using discontinuous approximations across elements but not artificial residuals, which hence requires more degrees of freedom than many other upwind-type schemes, such

as the SUPG method. Compared to many existing methods, our method has some unique features: (1)

Downloaded 08/29/18 to 128.175.16.112. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php A diﬀerent stabilization technique is used to avoid smearing the layers by over stabilization. (2) A set of strict assumptions for the convection coeﬃcient, in [9] and

other papers, is removed. (3) Error analysis has been signiﬁcantly simpliﬁed. Our error analysis is clean and short. (4) The discontinuity between elements makes it possible to use polytopal meshes (see section 2), which allow remarkable convenience in adaptive mesh generation and implementation; cf., e.g., [10, 13, 24, 25]. When shape-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 types of singularly perturbed convection-diffusion–reaction problems. Throughout this article, we use C for generic constants independent of , mesh size, and the solution to (1.1)–(1.2), which may not necessarily be the same at each

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

2. WG ﬁnite element schemes. For any given polyhedron D ⊆ Ω, we use the standard deﬁnition of Sobolev spaces Hs(D) with s ≥ 0. The associated inner s product, norm, and seminorms in H (D) are denoted by (·, ·)s,D, k · ks,D, and | · 0 |s,D, respectively. When s = 0, H (D) coincides with the space of square integrable 2 functions L (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 when D = Ω.

Let Th be a partition of the domain Ω consisting of elements which are closed and simply connected polygons in two dimensions or polyhedra in three dimensions; see 0 Figure 1. Let Eh be the set of all edges or ﬂat faces in Th, and Eh = Eh\∂Ω be the set of all interior edges or ﬂat faces. Denote by hT the diameter for every element

T ∈ Th and h = maxT ∈Th hT the mesh size for Th. We need some shape regularity assumptions for the partition Th described as below (cf. [37]). A1: Assume that there exist two positive constants % and % such that for every v e element T ∈ Th we have d d− 1 (2.1) %vhT ≤ |T |,%ehe ≤ |e|

for all edges or ﬂat faces of T . E F

xe n

D C A

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Fig. 1. Depiction of a shape-regular polygonal element ABCDEF A.

A2: Assume that there exists a positive constant κ such that for every element T ∈ 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 , and its height is proportional to hT with a proportionality constant σe bounded away from a fixed positive number σ∗ from below. In other words, the height ∗ of the pyramid is given by σehT such that σe ≥ σ > 0. The pyramid is also assumed to stand up above the base e in the sense that the angle between the vector xe − Ae, for any xe ∈ e, and the outward normal direction of e is π strictly acute by falling into an interval [0, θ0] with θ0 < 2 . A4: Assume that each T ∈ T has a circumscribed simplex S(T ) that is shape h regular 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, assume that each circumscribed simplex S(T ) intersects with only a fixed and small number of such simplices for all other elements T ∈ Th. For a given integer k ≥ 1, let V be the WG finite element space associated with h Th 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 0 = {v : v ∈ V , v = 0 on ∂Ω}, h h b

where Pk is the space of polynomials of total degree k or less. We would like to emphasize that any function v = {v0, vb} ∈ Vh has a single value vb on each edge e ∈ Eh, which is not necessarily the trace of v0 on e. The discontinuity between elements allows us to apply Pk spaces on polytopal elements, which brings convenience to many aspects in implementation. d For any v = {v0, vb}, a weak gradient ∇wv ∈ [Pk −1(T )] is deﬁned on T as the unique polynomial satisfying

d (2.5) (∇wv, τ)T = −(v0, ∇ · τ)T + hvb, τ · ni∂T ∀τ ∈ [Pk−1(T )] ,

where n is the unit outward normal vector to ∂T , and (·, ·)T and h·, ·i∂T are standard L2 inner products on T and ∂T , respectively. For any v = {v , v }, we deﬁne a weak 0 b divergence ∇w · (bv) ∈ Pk(T ) related to b on T as the unique polynomial satisfying

(2.6) (∇w · (bv), w)T = −(bv0, ∇w)T + hb · nvb , wi∂T ∀w ∈ Pk(T ).

We introduce four global projections Q0, Qb, Qh, and Qh. They are element- 2 wise deﬁned L projections detailed as follows. For each element T ∈ Th, Q0 : 2 2 2 L (T ) → Pk(T ) and Qb : L (e) → Pk(e) are the L projections onto the associated 2 d d 2 local polynomial spaces, h:[L (T )] → [ k−1(T )] is the L projection onto the local

Downloaded 08/29/18 to 128.175.16.112. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Q P weak gradient space. Finally, we deﬁne a projection operator Qhu = {Q0u, Qbu} ∈ Vh for the true solution u.

For simplicity, we adopt the following notations, Z X X (v, w)Th = (v, w)T = vwdx, T T ∈Th T ∈Th X X Z hv, wi = hv, wi = vwds. ∂Th ∂T ∂T T ∈Th T ∈Th

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

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

where

X s (v, w) = hb · n(v − v ), w − w i , c 0 b 0 b ∂+T T ∈T h X −1 sd(v, w) = hT hv0 − vb, w0 − wbi∂T , T ∈Th

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 u = {u , u } ∈ V 0 satisfying the h 0 b h following equation: 0 (2.8) a(uh, v) = (f, v0) ∀ v = {v0, vb } ∈ Vh .

Accordingly, we define an energy norm ||| · ||| in V 0 : for any v ∈ V 0, h h 2 2 X 2 X 1/2 2 (2.9) |||v||| = k∇wvkT + |b · n| (v0 − vb) + kv0k + sd(v, v). ∂T T ∈T T ∈T h h Remark 1. Compared to the WG scheme proposed in [9], our stabilizer for convection term s (·, ·) is simple and has upwinding flavor. Moreover, unlike in [2, 9], our c WG method does not make extra assumptions on the convection coefficient b. 3. Analysis. In this section, we will establish the well-posedness of the WG

method (2.8) and obtain an error estimate for the WG ﬁnite element approximation uh.

3.1. Well-posedness of the WG method and error equation. Veriﬁcation of continuity of the bilinear form (2.7) is straightforward. We next show the coercivity of the bilinear form. The following lemma is useful. 0 Lemma 3.1. For v, w ∈ Vh ,

Downloaded 08/29/18 to 128.175.16.112. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (∇w · (bv), w0)Th = (∇ · bv0, w0)Th − (v 0, ∇w · (bw))Th (3.1) − hb · n(v0 − vb), w 0 − wbi . ∂Th

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

(∇w · (bv), w0)T = −(bv0, ∇w0)T + hb · nvb, w0i h h ∂Th

= (∇ · bv0, w0)Th + (bw0, ∇v0)Th − hb · n(v0 − vb), w0i∂T h = (∇ · bv0, w0)T − (∇w · (bw), v0)T + hb · nwb, v0i h h ∂Th − hb · n(v0 − vb), w0i∂T h = (∇ · bv , w ) − (∇ · (bw), v ) 0 0 Th w 0 Th − hb · n(v0 − vb), w0 − wbi∂T , h which implies (3.1). Here we use the facts that v, w ∈ V 0 and hb · nv , w i = 0 to h b b ∂Th obtain the last equation above.

0 Lemma 3.2. For v ∈ Vh , then 2 (3.2) C|||v||| ≤ a(v, v).

Therefore, the WG formulation (2.8) has a unique solution. Proof. It follows from (3.1) that for any v ∈ V 0, h 1 1 (3.3) (∇w · (bv), v0)Th = (∇ · bv0, v0)Th − hb · n(v0 − vb), v0 − vbi . 2 2 ∂Th

Using (3.3), we have

1 a(v, v) = (∇ v, ∇ v) + c + ∇ · b v , v w w Th 2 0 0 T h 1 − hb · n(v0 − vb), v0 − vbi + sc(v, v) + sd(v, v) 2 ∂Th 1 2 2 X 1/2 ≥ (∇wv, ∇ww)Th + c0kv0k + |b · n | (v0 − vb) + sd(v, v) 2 ∂T T ∈Th 2 ≥ C|||v||| , which completes the proof.

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

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

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

where

` (u, v) = (u − Q u, b · ∇v ) , 1 0 0 Th `2(u, v) = hu − Qbu, b · n(v0 − vb)i∂T . h Proof. It follows from the deﬁnition of the weak divergence (2.6) that

(∇ · (bu), v ) = −(bu, ∇v ) + hb · nu, v i 0 Th 0 Th 0 ∂Th

= −(bQ0u, ∇v0)T − `1(u, v) + hb · nQbu, v0i h ∂Th − hb · nQ u, v i + hb · nu, v − v i b 0 ∂Th 0 b ∂Th

Downloaded 08/29/18 to 128.175.16.112. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php = (∇ · (bQ u), v ) − ` ( u, v) + ` (u, v), w h 0 Th 1 2 which implies (3.4).

0 Lemma 3.4. Let u be the solution of the problem (1.1)–(1.2). Then for v ∈ Vh ,

(3.5) −(∆u, v ) = (∇ Q u, ∇ v) − ` (u, v), 0 w h w Th a

where

`a(u, v) = h(∇u − h∇u) · n, v0 − vbi . Q ∂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 the following error equation. 0 0 Lemma 3.5. Let eh = Qhu − uh ∈ Vh . Then, for any v ∈ Vh we have

(3.6) a(eh, v) = `a(u, v) + `b(u, v) + `c(u, v) + sc(Q hu, v) + sd(Qhu, v). 0 Proof. Testing (1.1) by v = {v0, vb} ∈ Vh , we arrive at

(3.7) − (∆u, v0) + (∇ · (bu), v0) + (cu, v 0) = (f, v0). Using (3.4) and (3.5), (3.7) becomes

(∇ Q u, ∇ v) + (∇ · (bQ u), v ) + (cQ u, v ) w h w Th w h 0 Th 0 0 = (f, v0) + `a(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) + `a(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 proof of the lemma.

3.2. Error estimates. We will derive error estimates in this section. For any 1 function ϕ ∈ H (T ), the following trace inequality holds true: 2 −1 2 2 (3.9) kϕke ≤ C hT kϕkT + hT k∇ϕ kT .

Lemma 3.6. Let u be the solution of the problem (1.1)–(1.2) and Th be a ﬁnite element partition of Ω satisfying the shape regularity assumptions A1–A4. Then, the 2 L projections Q0 and Qh satisfy

X 2 2 2 2(s+1) 2 ku − Q0ukT + hT k∇(u − Q0u)kT ≤ Ch kuks+1, 0 ≤ s ≤ k, T ∈T h X 2 2 2 2s 2 k∇u − Qh∇ukT + hT |∇u − Qh∇u|1,T ≤ Ch kuks+1, 0 ≤ s ≤ k. T ∈Th

Proof. In order to avoid repetition, we refer the readers to the proof of [37, Lemma 4.1]. 0 Lemma 3.7. Let u be the solution of the problem (1.1)–(1.2). Then for v ∈ Vh , 1 k (3.10) |`a(u, v)| ≤ C 2 h |u|k+1||| v|||, k+ 1 (3.11) |` (u, v)| ≤ Ch 2 |u| |||v|||, b k+1 k+1 (3.12) |`c(u, v)| ≤ Ch |u|k+1||| v|||, 1

Downloaded 08/29/18 to 128.175.16.112. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php k+ (3.13) |sc(Qhu, v)| ≤ Ch 2 |u|k+1 |||v|||, 1 k (3.14) |sd(Qhu, v)| ≤ C 2 h |u|k+1||| v|||.

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

X |`a(u, v)| ≤ h(∇u − Qh∇u) · n, v0 − vbi∂T T ∈Th X ≤ C k∇u − ∇uk kv − v k Qh ∂T 0 b ∂T T ∈Th 1 1 ! 2 ! 2 X 2 X −1 2 ≤ C hT k∇u − Qh∇uk∂T h kv0 − vbk∂T T T ∈Th T ∈Th 1 ! 2 k X −1 2 ≤ Ch |u|k+1 hT kv0 − vbk∂T

T ∈Th k ≤ Ch |u|k+1|||v|||.

To prove (3.11), we need to bound ` (u, v) and ` (u, v) by the deﬁnition of ` (u, v). Let 1 2 b b 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,

|` (u, v)| = |(u − Q u, b · ∇v ) | = (u − Q u, (b − b) · ∇v ) 1 0 0 Th 0 0 Th k+1 ≤ Ch |u|k+1kv0k.

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

|` (u, v)| = hu − Q u, b · n(v − v )i 2 b 0 b ∂Th k+ 1 ≤ Ch 2 |u| |||v|||, k+1

where we use the fact ku − Qbuke ≤ ku − Q0uke. 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,

|s (Q u, v)| c h X ≤ hb · n(Q0u − Qbu), v0 − vbi ∂+T T ∈T h !1/2 !1/2 2 2 X 1/2 X 1/2 ≤ |b · n| (Q0u − u + u − Qbu) |b · n| (v0 − vb) ∂T ∂T T ∈Th T ∈Th 1/2 1/2 2 ! 2 ! X 1/2 X 1/2 ≤ C |b · n| (Q0u − u) |b · n| (v0 − vb) ∂T ∂T T ∈Th T ∈Th k+ 1 ≤ Ch 2 |u|k+1|||v|||.

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

0 Theorem 3.8. Let uh ∈ Vh 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

1/2 1/2 k (3.15) |||Qhu − uh||| ≤ C + h h |u|k+1.

Proof. It follows from (3.2) that

2 |||Qhu − uh||| ≤ Ca(eh, eh ).

Letting v = eh in (3.6) gives

a(e , e ) = ` (u, e ) + ` (u, e ) + ` (u, e ) + s ( Q u, e ) + s (Q u, e ). h h a h b h c h c h h d h h

Combining the two equations above with Lemma 3.7, we have

1/2 1/2 k |||Qhu − uh||| ≤ C + h h |u|k+1.

We have proved the theorem.

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

2 2 X 2 2 X 1/2 |||v||| = k∇wvkT + kv0k + |b · n| (v0 − vb) + sd(v, v), ∂T T ∈Th T ∈Th

and the SUPG norm |||·||| deﬁned as follows for a weak function v = {v , v } ∈ V , SUPG 0 b h

2 X X (3.16) |||v||| = k∇v k2 + kv k2 + δ kb · ∇v k2 . SUPG 0 T 0 T 0 T T ∈Th T ∈T h

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

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

! X X X (3.17) k∇v k2 ≤ C k∇ vk2 + h −1hv − v , v − v i . 0 T w T T 0 b 0 b ∂T T ∈Th T ∈Th T ∈Th

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

(∇wv, q)T = −(v0, ∇ · q)T + hvb, q · ni∂T = (∇v0 , q)T + hvb − v0, q · ni∂T .

By letting q = ∇v in the above identity we arrive at 0

Downloaded 08/29/18 to 128.175.16.112. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (∇wv, ∇v0)T = (∇v0, ∇v0)T + hvb − v 0, ∇v0 · ni∂T .

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

2 −1/2 k∇v0kT ≤ k∇wvkT k∇v0kT + ChT kv0 − vbk∂T k∇v0kT .

This leads to 1 k∇v k ≤ C k∇ vk2 + Ch−1kv − v k2 2 , 0 T w T T 0 b ∂T which completes the proof of the lemma.

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

! X 2 X 2 (3.18) k∇v0kT ≤ C k∇wvkT + sd(v, v) .

T ∈Th T ∈Th

Remark 2. It follows from (3.18) that the two norms |||·||| and |||·|||SUPG are similar, P 2 where the only difference is that the term δT kb·∇v0k in |||·||| is replaced T ∈Th T SUPG by P k|b·n|1/2(v −v )k2 in |||·|||. This difference reflects different “upwinding” T ∈Th 0 b ∂T strategies of the SUPG method and the WG method. The SUPG method adds a residual term to the continuous FEM to provide additional control on the convective derivative in the streamline direction. On the other hand, the term P k|b · T ∈Th 1/2 2 n| (v0 − vb)k∂T in ||| · ||| represents the upwinding strategy of using discontinuous approximation.

4. Numerical example. In this section, we test numerically the performance of the proposed WG FEM for solving boundary value problems (1.1)–(1.2). Examples

of different types have been tested to confirm the efficiency of the method. Numerical 2 errors will be measured in the L norm k · k and/or the energy norm ||| · |||. 4.1. Example 1: Smooth solution. This example is adopted from [2]. Let 2 T Ω = (0, 1) , b = (1, 1) , and c = 0. The source term f is chosen so that the exact solution is

u(x, y) = sin(2πx) sin(2πy ).

We computed approximate solutions to the boundary value problem for = 10−3 −9 and 10 to conﬁrm that the WG scheme is valid. Triangular meshes are used. Convergence plots for numerical errors for linear, quadratic, and cubic elements have been collected in Figure 2. It is observed that an optimal L2 estimate of O(hk+1) is k+1/2 obtained. On the other hand, |||Qhu − uh||| converges at O(h ), which matches the estimation in Theorem 3.8.

4.2. Example 2: Boundary layer. In this example, we examine the performance of the proposed WG method in the occurrence of the boundary layer. Let 2 T Ω = (0, 1) , b = (1, 1) , and c = 0 in (1.1). We choose f and an appropriate Dirichlet boundary condition so that the exact solution is πx πy (4.1) u(x, y) = sin + sin + e−1/ − e(x−1)( y−1)/ / 1 − e−1/ . 2 2

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

Downloaded 08/29/18 to 128.175.16.112. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php In particular, as shown in Table 1, the optimal convergence rate of O(hk+1/2) in Theorem 3.8 is conﬁrmed. For comparison, we measure also the errors of the WG

−3 −9 (a) kQ0u − u0k, = 10 (b) kQ0u − u0k, = 10

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

Fig. 2. Example 1. Convergence rates of kQ0u − u0k and of |||Qhu − uh|||of P1, P2, and P3 elements.

Table 1 Example 2. The numerical errors and their orders of convergence with P2 elements when = 10−9.

n n |||Qhu − uh||| h |||Qhu − uh||| h SUPG 1 0.0692366 0.1214335 2 0.0177851 2.26 0.0269782 2.50 3 0.0034200 2.57 0.0049298 2.65 4 0.0006320 2.54 0.0009027 2.55 5 0.0001142 2.52 0.0001631 2.52 6 0.0000204 2.51 0.0000292 2.51

ﬁnite element solutions in the standard SUPG energy norm [18] deﬁned by (3.16) (see also [32]). In Table 1, we present the numerical results with δT = hT in (3.16). It is observed that the convergence order of Qhu − uh in the SUPG norm is the same as that measured in the WG energy norm. We shall report that, as in other upwind schemes (see, e.g., [32]), our method

has disappointing performance inside the layer; cf. also Remark 1. In Table 2, it is observed that the numerical behavior of the method is poor in the intermediate

Downloaded 08/29/18 to 128.175.16.112. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php regimes (e.g., when = 10−3), which, however, is very good in the strongly advection- dominated cases (e.g., when = 10−9); see also Table 1. Similar phenomena in the

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

= 10−3 = 10−9 n n n kQ0u − u0k0 h kQ0u − u0k0 h |Q0u − u0|1,h h 1 0.0040535 0.08056691 0.3586332 2 0.0028010 0.60 0.00516027 4.57 0.0460337 3.41 3 0.0111441 0.00 0.00078110 2.95 0.0090941 2.53 4 0.0115248 0.00 0.00005975 3.87 0.0019822 2.29 5 0.0083321 0.48 0.00000750 3.06 0.0004856 2.07 6 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 1,024 elements, = 10−3.

(a) Numerical solution (b) Exact solution

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

DG method have been reported in the literature (see, e.g., [2, 19]). On the other −9 2 hand, when = 10 , optimal convergence of Q0u − u 0 in both the L norm and the H1 seminorm is obtained. −3 −9 The plots of uh for = 10 and = 10 over a mesh of 1,024 uniform elements are shown in Figures 3 and 4, respectively. In Figures 3 and 4, u0 is plotted for the numerical solutions uh = {u0, ub}. For the exact solution u, Q0u of Qhu = {Q0u, Qbu} is plotted and the average of Q0u and Qbu is plotted on ∂Ω.

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

Downloaded 08/29/18 to 128.175.16.112. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php α − x u(x, y) = 0.5x(1 − x)y(1 − y) 1 − tanh . γ

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

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

ε = 10−9 0 10

−1 10

−2 10

−3 10

−4 10 |||Q u−u ||| h h 3/2 −5 O(h ) 10 ||Q u−u || 0 0 O(h2) −6 10 −2 −1 0 10 10 10

Fig. 6. Example 3. Convergence rates of |||Qhu − uh||| and kQ0u − u0k of P1 element with −9 = 10 .

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 problem −9 with = 10 , α = 0.5, and γ = 0.05 are provided. When a P1 ﬁnite element space 3/2 is used, the numerical errors |||Qhu − uh||| and kQ0u − u0k converge in O(h ) and 2 O(h ), respectively. The interior layer can be accurately captured. 4.4. Example 4: Internal layer—discontinuous boundary conditions. 2 T Let Ω = (0, 1) , b = (1, 1) , and c = 0. We consider the following problem

(4.2) − ∆u + ∇ · (bu) + cu = 0 in Ω, ( 1 on {0} × (1/4, 1), (4.3) u = g = 0 elsewhere on ∂Ω .

Due to the discontinuities in boundary conditions, the solution of this problem is not in H1(Ω). Numerical oscillations occur near the interior layer caused by the joints of

Downloaded 08/29/18 to 128.175.16.112. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php the conﬂicting Dirichlet boundary conditions. This phenomenon has been reported in the literature for DG methods; see, e.g., [2, 19].

(a) u by element (b) u by element h P2 h P4

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

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

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 in Figure 7, in which oscillations are observed near the interior layer. We believe that a

better numerical approximation can be obtained if the Dirichlet boundary conditions are enforced weakly. That is, the FEM (2.8) is modiﬁed as ﬁnding uh = (u0, ub) ∈ Vh such that

(4.4) a(u , v) + hu , v i − h∇ u · n, v i = ( f, v ) + hg, v i ∀v ∈ V . h h b b ∂Ω w h b ∂Ω 0 h b ∂Ω h

The solutions to (4.4) using P2 and P4 elements are shown in Figure 8. Improved results 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 internal layer.

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

1 1 −∆u + ∇ · (bu) + cu = 0 in Ω = (0, 1)2 \ × 0, , 2 2 (4.5) 2 1 u = x(1 − x)y(1 − y) y − on ∂Ω, 2

−9 1 1 −3 where = 10 , b = ( −y, x− ), and c = 10 . k elements are used on rectangular Downloaded 08/29/18 to 128.175.16.112. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 2 2 P meshes. The WG solutions for k = 2 and k = 4 over a mesh of 256 elements are plotted in Figure 9. The numerical results demonstrate that the WG discretization is stable.

(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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