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Preprint 20 November 2020

A weighted and balanced FEM for singularly perturbed

reaction-diﬀusion problems

Niall Madden ∗

School of Mathematics, Statistics and Applied Mathematics,

National University of Ireland Galway, Ireland

and

Martin Stynes†

Applied and Computational Mathematics Division,

Beijing Computational Science Research Center, Beijing 100193, China

Abstract

A new ﬁnite element method is presented for a general class of singularly perturbed

reaction-diﬀusion problems −ε2∆u+bu =fposed on bounded domains Ω ⊂Rkfor k≥1,

with the Dirichlet boundary condition u= 0 on ∂Ω, where 0 < ε 1. The method is

shown to be quasioptimal (on arbitrary meshes and for arbitrary conforming ﬁnite element

spaces) with respect to a weighted norm that is known to be balanced when one has a typical

decomposition of the unknown solution into smooth and layer components. A robust (i.e.,

independent of ε) almost ﬁrst-order error bound for a particular FEM comprising piecewise

bilinears on a Shishkin mesh is proved in detail for the case where Ω is the unit square in

R2. Numerical results illustrate the performance of the method.

Keywords: Finite element method, balanced norm, quasioptimal MSC 2020 Classiﬁca-

tion: 65N30, 65N12

1 Introduction

Singularly perturbed diﬀerential equations of reaction-diﬀusion type have been extensively stud-

ied, as described in Linß (2010); Roos et al. (2008); Stynes and Stynes (2018). When a standard

Galerkin ﬁnite element method (FEM) is used to solve these problems, it is straightforward to

carry out the usual “energy norm” analysis, but a serious weakness of this measure of the error

is that when the singular perturbation parameter (i.e., the diﬀusion coeﬃcient in the PDE) is

very small, the energy norm of the error in the computed solution is essentially no stronger

than the L2norm of the error; that is, the H1component of the energy norm is typically much

smaller than its L2component. This drawback is discussed at length in Lin and Stynes (2012),

where the authors proposed the replacement of the energy norm by a stronger balanced norm

whose H1component is scaled to the correct size, so that both the H1and L2components of

the solution are O(1) when the singular perturbation parameter is small.

∗niall.madden@nuigalway.ie; ORCID 0000-0002-4327-4234

†m.stynes@csrc.ac.cn; ORCID 0000-0003-2085-7354; supported in part by the National Natural Science

Foundation of China under grant NSAF-U1930402.

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Lin and Stynes (2012) also proposed a new bilinear form and FEM that were designed to

facilitate analysis in their balanced norm. Subsequently, other authors derived new FEMs and

new analyses that yielded convergence in various balanced norms; see in particular Adler et al.

(2016, 2019); Cai and Ku (2020); Heuer and Karkulik (2017); Melenk and Xenophontos (2016);

Roos and Schopf (2015); Russell and Stynes (2019).

In the present paper, we consider a singularly perturbed reaction-diﬀusion problem that is

posed on an arbitrary bounded domain Ω ⊂Rkfor k≥1 and present a new and simple way of

constructing a FEM that is convergent in a weighted balanced norm. This new norm is stronger

than the standard energy norm because of the weighting that it includes.

Our key idea is to modify the standard Galerkin FEM by introducing a weight function

that was (essentially) already used in Adler et al. (2019), but unlike Adler et al. (2019) we do

not rewrite the reaction-diﬀusion problem as a system of equations. Furthermore, the analysis

of Adler et al. (2019) is for a problem posed on the unit square in R2, but the analysis in the

current paper permits a far more general class of domains.

Our method is simpler than any other FEM that is designed to yield convergence in a

balanced norm. It stands on a solid theoretical foundation: Theorem 5 shows that on an

arbitrary bounded domain, for arbitrary meshes and an arbitrary conforming FEM space, the

computed solution is quasioptimal with respect to our weighted norm. This norm can be shown

to be balanced for problems where one has a typical decomposition of the unknown solution into

smooth and layer components; see Remarks 2 and 7.

The structure of the paper is as follows. In Section 2 we state our reaction-diﬀusion problem

and construct a weighted norm and associated bilinear form for which we derive various fun-

damental properties. These results are used to construct our weighted FEM in Section 3, and

quasioptimality of the FEM solution is proved. Then in Section 4 we specialise this general the-

ory to the particular case where the domain is the unit square in 2D and the FEM uses bilinears

on a Shishkin mesh. Here a detailed error analysis leads to the optimal-order convergence result

of Theorem 12, where we show that (up to a log factor) the weighted FEM attains ﬁrst-order

convergence in our weighted norm. Finally, Section 5 presents numerical results to show the

performance of the weighted FEM.

2 The singularly perturbed reaction-diﬀusion problem

Let Ω be a bounded domain in Rk, where k≥1. Write ¯

Ω for the closure of Ω and ∂Ω for its

boundary. We shall discuss the elliptic boundary value problem

Lu := −ε2∆u+bu =fin Ω,(1a)

u= 0 on ∂Ω,(1b)

where the diﬀusion parameter satisﬁes 0 < ε ≤1, but the challenging and interesting case is

when ε1; then (1) is a singularly perturbed reaction-diﬀusion problem. Assume that the

reaction term satisﬁes b∈L∞(Ω) with b2

0< b(x)≤b2

1for almost all x∈Ω, where b0, b1are

positive constants. This assumption is usual in singularly perturbed problems of this type —

see Linß (2010); Roos et al. (2008); Stynes and Stynes (2018).

It is well known that for each f∈L2(Ω) the problem (1) has a unique solution u∈H2(Ω) ∩

H1

0(Ω) if Ω is convex or if ∂Ω is smooth, but throughout Sections 2 and 3 we assume only that

Ω is bounded.

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Preprint 20 November 2020

Notation. Let ωbe any measurable subset of ¯

Ω and let ∂Ω be its boundary. We use standard

notation for the Sobolev spaces Wm,p(ω) and their associated norms k · km,p,ω and seminorms

|·|m,p,ω. If m= 0, we simply write k·kp,ω; while if p= 2 we set Hm(ω) = Wm,2(ω) and

k · km,ω =k · km,2,ω. As usual, H1

0(ω) = {v∈H1(ω) : v|∂Ω= 0}, where v|∂Ωis the trace of v

on ∂Ω, and L2(ω) = H0(ω).

Throughout this paper, the letter C(with or without subscripts) will denote a generic pos-

itive constant that may stand for diﬀerent values in diﬀerent places but is independent of the

parameter εand of the mesh diameter.

2.1 Weighted norm

To solve (1), we shall apply a FEM that uses standard trial and test functions on general

meshes; its distinguishing feature is that it incorporates a special weight in the bilinear form

and its associated norm, where the weight is chosen in such a way that the norm is balanced

(i.e., the H1(Ω) and L2(Ω) weighted components of our norm are commensurable for typical

solutions of (1)); see Remarks 2 and 7.

For each x∈Ω, set

d(x) = min{|x−z|:z∈∂Ω},

where |x−z|denotes the Euclidean distance from xto z. That is, d(x) is the Euclidean distance

from xto the boundary of Ω.

For any y∈Ω, choose z∈∂Ω such that d(y) = |y−z|. Then d(x)≤ |x−z|≤|x−y|+d(y),

by a triangle inequality. Similarly d(y)≤ |y−x|+d(x). Hence

|d(x)−d(y)|≤|x−y|for all x, y ∈Ω.(2)

(This proof of (2) comes from Gilbarg and Trudinger (2001, p.354).) The inequality (2) says that

d(·) is uniformly Lipschitz continuous on ¯

Ω. Then Rademacher’s Theorem (Evans, 2010, p.296)

guarantees that the function d(·) is diﬀerentiable almost everywhere in Ω, and a fortiori (2)

clearly implies that

|∇d(x)| ≤ 1 almost everywhere.(3)

Deﬁne our weight function βby

β(x) = 1 + 1

εexp −γd(x)

εfor x∈Ω,(4)

where the positive constant γis (for the moment) arbitrary. Then

∇β(x) = −γ∇d(x)

ε2exp −γd(x)

εalmost everywhere in Ω.

Recalling (3), we get

|∇β(x)| ≤ γ

ε2exp −γd(x)

ε≤γ

εβ(x) almost everywhere in Ω.(5)

The weight function βdeﬁned in (4) is a generalisation of the weight function used in Adler

et al. (2019) for a problem posed on the unit square in R2.

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Preprint 20 November 2020

Let (·,·) denote the scalar and vector inner products in L2(Ω). Write k·kfor the norm

associated with this inner product. For each v∈L2(Ω) and w∈H1(Ω), deﬁne

kvkβ:= (βv, v)1/2and |||w|||β:= ε2k∇wk2

β+kwk2

β1/2.

This norm is balanced (i.e., its components ε2(β∇u, ∇u) and kuk2

βare both O(1)) for a typical

solution uof (1); see Remarks 2 and 7.

Remark 1 (Comparison with other balanced norms used to solve (1)).In Adler et al. (2019) the

balanced norm used comprises the above norm |||w|||βwith some extra terms added. The balanced

norm used in Cai and Ku (2020); Melenk and Xenophontos (2016); Roos and Schopf (2015);

Russell and Stynes (2019) is kwkε:= εk∇wk2+kwk21/2. In Adler et al. (2016); Heuer and

Karkulik (2017); Lin and Stynes (2012) extra terms are added to kwkεto form a balanced norm.

Of course, all these norms are stronger than the standard “energy norm” ε2k∇wk2+kwk21/2

that is often associated with (1).

Remark 2 (Heuristic justiﬁcation that |||· |||βis balanced).Given suﬃcient information about

the data of (1), a standard technique due to Shishkin enables us to decompose its solution as

u=v+w, where vsatisﬁes Lv =fon Ωwith all low-order derivatives of vbounded independently

of ε, while Lw = 0 on Ωwith w=−von ∂Ω, so wis the boundary layer component of u. This is

done for example in Clavero et al. (2005), where Ω⊂R2is the unit square. Using the properties

of vand w, one ﬁnds that typically |||v|||β∼ kvkβ=O(1) and |||w|||β∼εk∇wkβ=O(1), so

each component of |||u|||βis O(1), demonstrating that ||| · |||βis balanced.

2.2 Weighted bilinear form

For all v, w ∈H1

0(Ω), deﬁne the bilinear form

Bβ(v, w) := (ε2∇v , ∇(βw)) + (bv, β w),

We prove two fundamental properties of Bβ(·,·).

Lemma 3 (coercivity of Bβ(·,·)).Assume that 0< γ ≤b0. Then for the weight function β

speciﬁed in (4), one has

Bβ(v, v)> C1|||v|||2

βfor all v∈H1

0(Ω),

where C1:= min{1/2, b2

0/2}.

Proof. Let v∈H1

0(Ω). Then

Bβ(v, v)=(ε2∇v, ∇(βv)) + (bv, βv)

≥ε2(∇v, β∇v+v∇β)+(b0v, βv)

=ε2k∇vk2

β+ (ε2∇v, v∇β) + b2

0kvk2

β.(6)

Recalling (5), we have

(ε2∇v, v∇β)≤(ε|∇v|,|v|γβ)

≤εk∇vkβγkvkβ

≤ε2

2k∇vk2

β+γ2

2kvk2

β,

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Preprint 20 November 2020

by the Cauchy-Schwarz and Young inequalities. Inserting this bound into (6), one obtains

Bβ(v, v)≥ε2

2k∇vk2

β+b2

0−γ2

2kvk2

β.

The lemma follows, since γ≤b0.

Lemma 4 (boundedness of Bβ(·,·)).For the weight function βspeciﬁed in (4), one has

|Bβ(v, w)| ≤ C2|||v|||β|||w|||βfor all v, w ∈H1

0(Ω),

where C2:= max{1, γ, b2

1}.

Proof. Let v, w ∈H1

0(Ω). Then

Bβ(v, w) = (ε2∇v , ∇(βw)) + (bv, β w)=(ε2β∇v, ∇w)+(ε2∇v, w∇β)+(bv, β w).(7)

Now (5) yields

|(ε2∇v, w∇β)| ≤ γε(|∇v|,|w|β)≤γεk∇vkβkwkβ

by a Cauchy-Schwarz inequality. Applying two more Cauchy-Schwarz inequalities to (7), we get

|Bβ(v, w)| ≤ ε2k∇vkβk∇wkβ+γεk∇vkβkwkβ+b2

1kvkβkwkβ.

The desired result follows.

From now on, we assume that f∈H−1(Ω) := H1

0(Ω)0. One can then deﬁne a weak solution

u∈H1

0(Ω) of (1) by requiring it to satisfy

Bβ(u, v)=(f , βv) for all v∈H1

0(Ω),(8)

where βis deﬁned in (4) and the constant γsatisﬁes 0 < γ ≤b0. By the Lax-Milgram theorem,

invoking Lemmas 3 and 4, the problem (8) has a unique solution u∈H1

0(Ω). Any strong solution

u∈H2(¯

Ω) ∩H1

0(Ω) of the original problem (1) is also a weak solution of (8), as can be seen by

multiplying (1a) by βv then integrating by parts.

3 The weighted ﬁnite element method

Let Ωhbe an arbitrary mesh on ¯

Ω. Let Vh⊂H1

0(Ω) be a conforming ﬁnite element space deﬁned

on this mesh. Suppose that 0 < γ ≤b0. From the Lax-Milgram theorem and Lemmas 3 and 4,

there is a unique uh∈Vhsuch that

Bβ(uh, vh)=(f, β vh) for all vh∈Vh.(9)

Combining (8) and (9) gives the Galerkin orthogonality property

Bβ(u−uh, vh) = 0 for all vh∈Vh.(10)

Theorem 5 (quasioptimal FEM error bound).Let Ωbe a bounded subset of Rkfor some k≥1.

Let Ωhbe an arbitrary mesh on ¯

Ω, and Vh⊂H1

0(Ω) a ﬁnite element space deﬁned on this mesh.

Choose γ∈Rto satisfy 0< γ ≤b0. Let ube the weak solution of (8) and uhthe solution of (9).

Then one has

|||u−uh|||β≤C2

C1

inf

wh∈Vh|||u−wh|||β.

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Preprint 20 November 2020

Proof. Let wh∈Vhbe arbitrary. Invoking Lemma 3, then equation (10), then Lemma 4, we get

C1|||u−uh|||2

β≤Bβ(u−uh, u −uh)

=Bβ(u−uh, u −wh)

≤C2|||u−uh|||β|||u−wh|||β.

Hence |||u−uh|||β≤(C2/C1)|||u−wh|||β. As wh∈Vhwas arbitrary, we are done.

The analysis up to this point is for a general bounded domain, a general mesh, and a general

conforming ﬁnite element space Vh. This generality cannot however yield a bound in Theorem 5

that will ensure that |||u−uh|||βis small; to get this desirable outcome, one must tailor Vhor

the mesh to the singularly perturbed nature of the problem. In Section 4 we show how this is

done for a speciﬁc domain by a suitable choice of mesh.

4 Reaction-diﬀusion problem on the unit square in R2

We now specialise the theory of Section 2 to a 2D reaction-diﬀusion problem that has been

considered in many balanced-norm papers, including Adler et al. (2016); Heuer and Karkulik

(2017); Lin and Stynes (2012).

During Section 4 we take Ω = (0,1)2, the unit square in R2. Assume that f, b ∈C0,α(¯

Ω),

where we use the standard notation for H¨older spaces. Assume the corner compatibility condi-

tions

f(0,0) = f(1,0) = f(0,1) = f(1,1) = 0.

Then (1) has a unique solution u∈C2,α(¯

Ω); see, e.g., Han and Kellogg (1990). Furthermore,

this solution has typically an exponential boundary layer in a neighbourhood of ∂Ω of width

O(ε|ln ε|); see Lemma 6 below for more details.

The 4 sides of Ω will be denoted by

Γ1:= {(x, 0)|0≤x≤1},Γ2:= {(0, y)|0≤y≤1},

Γ3:= {(x, 1)|0≤x≤1},Γ4:= {(1, y)|0≤y≤1}.

The corners of this domain are z1:= (0,0), z2:= (1,0), z3:= (1,1), z4:= (0,1).

In Clavero et al. (2005, Theorem 2.2) and Liu et al. (2009, Lemma 1.2) the solution u

of (1) with Ω = (0,1)2is decomposed into smooth and layer components, and bounds are

derived on certain derivatives of these components. The analysis in these papers makes stronger

assumptions than ours, but an inspection of their arguments shows however that, under our

assumptions on the data, one obtains the following result, where for brevity the derivative

∂m+ng/∂mx∂nyof any function gis written as Dm

xDn

yg.

Lemma 6. The solution uof (1) can be decomposed as

u=v+

4

X

k=1

wk+

4

X

k=1

zk,(11)

where each wkis a layer associated with the edge Γkand each zkis a layer associated with the

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Preprint 20 November 2020

corner ck. There exists a constant Csuch that for all (x, y)∈¯

Ωand 0≤m+n≤2one has

kDm

xDn

yvk0,∞,¯

Ω≤C, (12a)

Dm

xDn

yw1(x, y)≤Cε−ne−b0y /ε,(12b)

Dm

xDn

yz1(x, y)≤Cε−m−ne−b0(x+y)/ε .(12c)

Bounds for w2, w3and w4that are analogous to (12b) and bounds for z2, z3and z4that are

analogous to (12c) also hold.

Remark 7. For a solution uthat enjoys the properties described in Lemma 6, one can verify

that for any constant γ > 0, one has εk∇ukβ=O(1) and kukβ=O(1). That is, the norm

||| · |||βis balanced for this problem.

4.1 The Shishkin mesh

To solve the problem numerically, we shall use a piecewise-uniform Shishkin mesh. These meshes

are a popular tool in the numerical solution of problems such as (1). For an introduction to

their properties and usage, see Linß (2010); Roos et al. (2008); Stynes and Stynes (2018).

Let Nbe an even positive integer. The mesh will use Nmesh intervals in each coordinate

direction. The mesh transition parameter λwill specify where the mesh changes from coarse to

ﬁne; we deﬁne it by

λ= min 1

4,σε ln N

b0.(13)

In this formula, the constant σwill be chosen later to facilitate our numerical analysis and is

then used in the implementation of the ﬁnite element method.

Then, without loss of generality, one can assume that Nis so large that (13) simpliﬁes to

λ=σεb−1

0ln N. (14)

Partition Ω as follows (see Figure 1): ¯

Ω=Ω11 ∪Ω21 ∪Ω12 ∪Ω22, where

Ω11 = [λ, 1−λ]×[λ, 1−λ],Ω21 = ([0, λ]∪[1 −λ, 1]) ×[λ, 1−λ],

Ω12 = [λ, 1−λ]×([0, λ]∪[1 −λ, 1]),

Ω22 = ([0, λ]×([0, λ]∪[1 −λ, 1])) ∪([1 −λ, 1] ×([0, λ]∪[1 −λ, 1])).

Divide each of the x-intervals [0, λ] and [1 −λ, 1] into N/4 equidistant subintervals and divide

[λ, 1−λ] into N/2 equidistant subintervals. This gives a coarse mesh on [λ, 1−λ] and a ﬁne mesh

on [0, λ]∪[1 −λ, 1]. Divide the y-interval [0,1] in the same way. Then the ﬁnal 2-dimensional

mesh is a tensor product of these 1-dimensional Shishkin meshes; see Figure 1, where N= 8.

An explicit description of the mesh follows: one has 0 = x0< x1<·· · < xN= 1 and

0 = y0< y1<·· · < yN= 1, with mesh sizes hi:= xi−xi−1and kj:= yj−yj−1that are deﬁned

by

hi=(h:= 4λN−1for i= 0, . . . , N/4 and i= 3N/4+1, . . . , N,

H:= 2(1 −2λ)N−1for i=N/4+1,...,3N/4,

(15a)

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r r

r

r

λ

1−λ

λ1−λ

Ω22

Ω22

Ω22

Ω22

Ω12

Ω12

Ω21 Ω21

Ω11

r r

r

r

Figure 1: Shishkin mesh for reaction-diﬀusion

and

kj=(hfor j= 0, . . . , N/4 and j= 3N/4+1, . . . , N,

Hfor j=N/4+1,...,3N/4.

(15b)

The mesh divides Ω into a set TN,N of mesh rectangles Rwhose sides are parallel to the

axes—see Figure 1. The mesh is coarse on Ω11, coarse/ﬁne on Ω21 ∪Ω12, and ﬁne on Ω22 .

The mesh is quasiuniform on Ω11 and its diameter dthere satisﬁes √2/N ≤d≤2√2/N; on

Ω12 ∪Ω21, each mesh rectangle has dimensions O(N−1) by O(εN−1ln N); and on Ω22 each

rectangle is O(εN−1ln N) by O(εN−1ln N). These dimensions will be used in the error analysis

of our ﬁnite element method.

4.2 Bilinear FEM on Shishkin mesh

We assume henceforth that the user-chosen constant γsatisﬁes 0 < γ ≤b0, so Theorem 5 is

valid.

For our ﬁnite element method, choose the ﬁnite space Vh⊂C(¯

Ω) ∩H1

0(Ω) to comprise

piecewise bilinears on the Shishkin mesh of Section 4.1. Given any function g∈C(¯

Ω) ∩H1

0(Ω),

we write gIfor the nodal interpolant of gfrom Vh.

The following interpolation error bound is suitable for the highly anisotropic Shishkin mesh.

Lemma 8. Apel (1999, Theorem 2.7) Let Rbe any Shishkin mesh rectangle with dimensions

hx×ky. Let φ∈H2(R). Then its bilinear nodal interpolant φIsatisﬁes the bounds

kφ−φIk∞,R ≤Ch2

xkφxxk∞,R +k2

ykφyy k∞,R,

k(φ−φI)xk∞,R ≤C(hxkφxxk∞,R +kykφxy k∞,R),

k(φ−φI)yk∞,R ≤C(hxkφxyk∞,R +kykφyyk∞,R ),

where the constant Cis independent of φ, hxand ky.

Theorem 5 gives us

|||u−uh|||β≤C2

C1|||u−uI|||β.(16)

We shall bound the right-hand side of (16) by using the decomposition of ufrom Lemma 6,

comparing each component there with its interpolant from Vh.

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Lemma 9. For the component vin Lemma 6, one has

|||v−vI|||β≤CεN−1+N−2.

Proof. Lemmas 6 and 8 yield kv−vIk∞,Ω≤CN−2|v|2,∞,Ω≤CN −2. Hence

kv−vIkβ=ZΩ

β(v−vI)21/2

≤CN−2ZΩ

β1/2

≤CN−2.

By a similar argument, again using Lemmas 6 and 8, one has k∇(v−vI)k∞,Ω≤CN−1|v|2,∞,Ω≤

CN−1and hence k∇(v−vI)kβ≤C N −1. To ﬁnish the proof, recall the deﬁnition of ||| ·|||β.

For any measurable ω⊂Ω, and each v∈L2(ω) and w∈H1(ω), deﬁne

kvkβ,ω := Zω

βv21/2

and |||w|||β,ω := ε2Zω

β|∇w|2+kwk2

β,ω 1/2

.

Lemma 10. For the components w1, w2, w3, w4of Lemma 6, one has

|||wj−wI

j|||β≤CN−1ln N+N−σfor j= 1,2,3,4.

Proof. We give the proof only for w1, as the other wjare similar.

Set Ωy={(x, y)∈Ω : y≥λ}. By Lemma 6, one has

Dm

xDn

yw1(x, y)≤Cε−nN−σfor 0 ≤m+n≤2 and (x, y)∈Ωy.

Hence,

kw1−wI

1k∞,Ωy≤ kw1k∞,Ωy+kwI

1k∞,Ωy= 2kw1k∞,Ωy≤CN−σ

and

k∇(w1−wI

1)k∞,Ωy≤ k∇w1k∞,Ωy+k∇wI

1k∞,Ωy≤2k∇w1k∞,Ωy≤Cε−1N−σ.

It now follows, as in the proof of Lemma 9, that

|||w1−wI

1|||β,Ωy≤CN−σ.(17)

On Ω \Ωyinvoke Lemma 8 and (15) to get

kw1−wI

1k∞,Ω\Ωy≤CH2k(w1)xxk∞,Ω\Ωy+h2k(w1)yyk∞,Ω\Ωy

≤CN−2+ (εN−1ln N)2ε−2

=C(N−1ln N)2

and

k∇(w1−wI

1)k∞,Ω\Ωy≤CHk(w1)xxk∞,Ω\Ωy+ (h+H)k(w1)xy k∞,Ω\Ωy

+hk(w1)yy k∞,Ω\Ωy

≤CN−1(1 + ε−1)+(εN−1ln N)(ε−1+ε−2)

=Cε−1(N−1ln N),

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where we used Lemma 6 to bound the derivatives of w1. From these estimates it follows, as in

the proof of Lemma 9, that

|||w1−wI

1|||β,Ω\Ωy≤CN−1ln N. (18)

Putting together (17) and (18) completes the proof.

Next we prove the corresponding result for the components zjof u; the proof resembles the

proof of Lemma 10 but there are some diﬀerences.

Lemma 11. For the components z1, z2, z3, z4in Lemma 6, one has

|||zj−zI

j|||β≤CN−1ln N+N−σfor j= 1,2,3,4.

Proof. We give the proof only for z1, as the other zjare similar.

Set Ωxy ={(x, y)∈Ω : x≥λor y≥λ}. By Lemma 6, one has

Dm

xDn

yw1(x, y)≤Cε−m−nN−σfor 0 ≤m+n≤2 and (x, y)∈Ωxy.

Hence,

kz1−zI

1k∞,Ωy≤ kz1k∞,Ωy+kzI

1k∞,Ωy= 2kz1k∞,Ωy≤CN−σ

and

k∇(z1−zI

1)k∞,Ωy≤ k∇z1k∞,Ωy+k∇zI

1k∞,Ωy≤2k∇z1k∞,Ωy≤Cε−1N−σ.

It now follows, as in the proof of Lemma 9, that

|||z1−zI

1|||β,Ωy≤CN−σ.(19)

On Ω \Ωxy invoke Lemma 8 and (15) to get

kz1−zI

1k∞,Ω\Ωy≤Ch2k(z1)xxk∞,Ω\Ωy+h2k(z1)yyk∞,Ω\Ωy

≤C(εN−1ln N)2ε−2

=C(N−1ln N)2

and

k∇(z1−zI

1)k∞,Ω\Ωy≤Chk(z1)xxk∞,Ω\Ωy+hk(z1)xy k∞,Ω\Ωy

+hk(z1)yy k∞,Ω\Ωy

≤C(εN−1ln N)(1 + ε−1+ε−2)

=Cε−1(N−1ln N),

where we used Lemma 6 to bound the derivatives of z1. From these estimates it follows, as in

the proof of Lemma 9, that

|||z1−zI

1|||β,Ω\Ωy≤CN−1ln N. (20)

Combine (19) and (20) to ﬁnish the argument.

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We can now give a precise error estimate for our ﬁnite element method.

Theorem 12. Let ube the solution of (1) and uhthe solution of (9), where the ﬁnite element

method uses piecewise bilinears on the Shishkin mesh. Then there exists a constant Csuch that

|||u−uh|||β≤CN−1ln N+N−σ.

With the choice σ= 1 in the deﬁnition (14) of the Shishkin mesh transition parameter, this

bound becomes

|||u−uh|||β≤CN−1ln N.

Proof. Lemmas 9, 10 and 11 and the decomposition of Lemma 6 yield

|||u−uI|||β≤CN−1ln N+N−σ.

The result then follows from Lemma 5.

Remark 13. One could instead use a FEM space that comprises polynomials of higher degree

and carry out a similar error analysis to obtain a higher order of convergence in Theorem 12,

but this necessitates imposing more conditions on the data of (1) so that more derivatives of u

are bounded in Lemma 6, as in Clavero et al. (2005); Liu et al. (2009).

Remark 14. Franz and Roos (2014) extend the reaction-diﬀusion analysis of Roos and Schopf

(2015) to the convection-diﬀusion problem

−ε∆v+avx+bv =fon Ω = (0,1)2, v = 0 on ∂Ω,(21)

where the functions aand bare positive. (Here, as is customary, we have replaced the ε2

coeﬃcient of (1a) by εsince this is a convection-diﬀusion problem.) Typical solutions of this

problem exhibit an exponential layer along the side x= 1 of Ω, and parabolic layers along the

sides y= 0 and y= 1. The associated energy norm ε|∇v|2+kvk2

01/2is correctly balanced

for the exponential layer, but is unbalanced for the weaker parabolic layers, for which it reduces

essentially to the L2(Ω) norm when ε1. In Franz and Roos (2014) an FEM comprising

streamline-diﬀusion (SDFEM) in the x-direction and standard Galerkin in the y-direction is

used and is shown to converge in the balanced norm ε|vx|2+ε1/2|vy|2+kvk2

01/2on a Shishkin

mesh that is appropriate for this problem.

One can carry out an analogous construction and analysis in our setting: working with

piecewise bilinears on a Shishkin mesh like that of Franz and Roos (2014), construct an FEM

that uses SDFEM in the x-direction (note that we can use the standard SDFEM, unlike the

special variant of SDFEM that is used in Franz and Roos (2014)) and replaces (4) by the weight

function

ˆ

β(x, y) := 1 + 1

ε1/2exp −γ[1/2− |y−1/2|]

ε1/2for (x, y)∈Ω.

This weight is a function of yonly and is large inside the parabolic layers. One can prove

convergence on the Shishkin mesh in the weighted balanced norm

hε|vx|2+εˆ

β|vy|2+kvk2

0i1/2.(22)

We do not give details here as they are lengthy and require no new idea beyond a synthesis of

our earlier analysis and that of Franz and Roos (2014).

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Table 1: |||u−uh|||βwhere uis as in (24) and uhthe solution of (9) on a Shishkin mesh

ε N = 32 N= 64 N= 128 N= 256 N= 512

1 4.303e-03 2.151e-03 1.076e-03 5.378e-04 2.689e-04

10−12.975e-02 1.787e-02 1.043e-02 5.320e-03 2.660e-03

10−23.322e-02 1.985e-02 1.157e-02 6.610e-03 3.718e-03

10−33.360e-02 2.007e-02 1.169e-02 6.675e-03 3.754e-03

10−43.358e-02 2.008e-02 1.170e-02 6.682e-03 3.758e-03

10−53.366e-02 2.009e-02 1.170e-02 6.682e-03 3.758e-03

10−63.366e-02 2.010e-02 1.170e-02 6.683e-03 3.758e-03

10−73.366e-02 2.010e-02 1.170e-02 6.683e-03 3.758e-03

10−83.366e-02 2.010e-02 1.170e-02 6.683e-03 3.758e-03

5 Numerical results

Our test problem is

−ε2∆u+u=fin Ω = (0,1)2, u = 0 on ∂Ω,(23)

where fchosen so that

u(x, y) = cos π

2x−e−x/ε −e−1/ε

1−e−1/ε !1−y−e−y/ε −e−1/ε

1−2−1/ε .(24)

This problem is taken from Kopteva (2008) and is widely used in the literature (see, e.g., Adler

et al. (2016)). Its solution exhibits only two layers, which are near x= 0 and y= 0. Nonetheless,

the bounds presented in Theorem 6 hold and are sharp for v,w1,w2, and z1. Thus, the example

is suﬃciently typical to verify the results of Theorem 12.

Since this problem has layers along only two edges, we modify the Shishkin mesh of Sec-

tion 4.1 so that it is a tensor product of two one-dimensional meshes with N/2 equidistant

subintervals on each of [0, λ] and [λ, 1]. In our experiments we have taken γ= 0.98 in (4), and

b0= 0.99 and σ= 1 in (13).

Our results are computed using Firedrake; see Rathgeber et al. (2016). All results presented

here are for bilinear elements; consistent results were obtained using biquadratic elements (see

Remark 13).

In Table 1 we present the errors in the solutions computed using our proposed method.

Observe that, for suﬃciently small ε, the errors are independent of ε, and converge at a rate

that is O(N−1ln N), verifying Theorem 12. This contrasts with the observed results for the

classical Galerkin method (i.e., β(·)≡1) for this problem on this Shishkin mesh, where the

computed errors in the standard energy norm scale like ε1/2N−1ln N; see, e.g., (Liu et al., 2009,

Table 1).

Although a discrete maximum norm error analysis is beyond the scope of this paper, in

Table 2 we show the maximum error observed at the mesh points. One sees again that, for

suﬃciently small ε, the error is independent of εand converges at a rate that is O(N−1). We

remark that choosing σ= 2 in (13) improves the experimental convergence rate to O(N−2ln2N),

though we do not show these results here.

Finally, we mention that when a solution ˜uhis computed using the classical Galerkin FEM,

our numerical results reveal that |||u−˜uh|||βclosely matches the values of |||u−uh|||βstated

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Table 2: |||u−uh|||∞,T N,N for the weighted method on a Shishkin mesh

ε N = 32 N= 64 N= 128 N= 256 N= 512

1 3.901e-05 9.673e-06 2.372e-06 5.699e-07 1.313e-07

10−12.678e-03 9.603e-04 3.283e-04 8.664e-05 2.231e-05

10−21.580e-02 4.857e-03 8.965e-04 1.549e-04 4.249e-05

10−33.180e-02 1.445e-02 5.990e-03 2.090e-03 7.605e-04

10−43.406e-02 1.670e-02 8.142e-03 3.912e-03 1.818e-03

10−52.553e-02 1.398e-02 7.951e-03 4.132e-03 2.041e-03

10−62.526e-02 1.306e-02 6.694e-03 3.413e-03 1.753e-03

10−72.526e-02 1.306e-02 6.697e-03 3.412e-03 1.729e-03

10−82.526e-02 1.306e-02 6.697e-03 3.412e-03 1.729e-03

in Table 1. This is curious, and worthy of further investigation, particularly since our exper-

iments also suggest that when errors are measured in the discrete maximum norm, then our

weighted method is more accurate than the classical Galerkin FEM by more than a factor of

two.

Remark 15. For the problem of Remark 14, we have veriﬁed experimentally the accuracy of

the numerical method described in that remark, using the standard SDFEM with its stabilisation

parameter chosen as in Linß (2005). Here we chose fin (21) such that the solution has parabolic

layers along y= 0 and y= 1, but no exponential layer at the outﬂow boundary x= 1, so that

the error in the parabolic layer dominates. Our results in the weighted norm (22) show almost

ﬁrst-order convergence, independently of ε.

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