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Abstract

A new finite element method is presented for a general class of singularly perturbed reaction-diffusion problems $-\eps^2\Delta u +bu=f$ posed on bounded domains $\Omega\subset \mathbb{R}^k$ for $k\ge 1$, with the Dirichlet boundary condition $u=0$ on~$\partial\Om$, where $0 <\eps \ll 1$. 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 $\eps$) almost first-order error bound for a particular FEM comprising piecewise bilinears on a Shishkin mesh is proved in detail for the case where $\Omega$ is the unit square in $\mathbb{R}^2$. Numerical results illustrate the performance of the method.
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 ε2u+bu =fposed on bounded domains Ω Rkfor k1,
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 k1 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 k1. Write ¯
Ω for the closure of Ω and Ω for its
boundary. We shall discuss the elliptic boundary value problem
Lu := ε2u+bu =fin Ω,(1a)
u= 0 on ,(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 bL(Ω) with b2
0< b(x)b2
1for almost all xΩ, 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 fL2(Ω) the problem (1) has a unique solution uH2(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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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. As usual, H1
0(ω) = {vH1(ω) : v|= 0}, where v|is 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 xΩ, set
d(x) = min{|xz|:z},
where |xz|denotes the Euclidean distance from xto z. That is, d(x) is the Euclidean distance
from xto the boundary of Ω.
For any yΩ, choose zΩ such that d(y) = |yz|. Then d(x)≤ |xz|≤|xy|+d(y),
by a triangle inequality. Similarly d(y)≤ |yx|+d(x). Hence
|d(x)d(y)|≤|xy|for all x, y .(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,(4)
where the positive constant γis (for the moment) arbitrary. Then
β(x) = γd(x)
ε2exp γd(x)
εalmost everywhere in Ω.
Recalling (3), we get
|∇β(x)| ≤ γ
ε2exp γd(x)
εγ
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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Let (·,·) denote the scalar and vector inner products in L2(Ω). Write k·kfor the norm
associated with this inner product. For each vL2(Ω) and wH1(Ω), define
kvkβ:= (βv, v)1/2and |||w|||β:= ε2k∇wk2
β+kwk2
β1/2.
This norm is balanced (i.e., its components ε2(βu, 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 with all low-order derivatives of vbounded independently
of ε, while Lw = 0 on with w=von , so wis the boundary layer component of u. This is
done for example in Clavero et al. (2005), where R2is the unit square. Using the properties
of vand w, one finds that typically |||v|||β∼ kvkβ=O(1) and |||w|||βεk∇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) := (ε2v , (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 vH1
0(Ω),
where C1:= min{1/2, b2
0/2}.
Proof. Let vH1
0(Ω). Then
Bβ(v, v)=(ε2v, (βv)) + (bv, βv)
ε2(v, βv+vβ)+(b0v, βv)
=ε2k∇vk2
β+ (ε2v, vβ) + b2
0kvk2
β.(6)
Recalling (5), we have
(ε2v, vβ)(ε|∇v|,|v|γβ)
εk∇vkβγkvkβ
ε2
2k∇vk2
β+γ2
2kvk2
β,
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by the Cauchy-Schwarz and Young inequalities. Inserting this bound into (6), one obtains
Bβ(v, v)ε2
2k∇vk2
β+b2
0γ2
2kvk2
β.
The lemma follows, since γb0.
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) = (ε2v , (βw)) + (bv, β w)=(ε2βv, w)+(ε2v, wβ)+(bv, β w).(7)
Now (5) yields
|(ε2v, wβ)| ≤ 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 fH1(Ω) := H1
0(Ω)0. One can then define a weak solution
uH1
0(Ω) of (1) by requiring it to satisfy
Bβ(u, v)=(f , βv) for all vH1
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 uH1
0(Ω). Any strong solution
uH2(¯
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 VhH1
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 uhVhsuch that
Bβ(uh, vh)=(f, β vh) for all vhVh.(9)
Combining (8) and (9) gives the Galerkin orthogonality property
Bβ(uuh, vh) = 0 for all vhVh.(10)
Theorem 5 (quasioptimal FEM error bound).Let be a bounded subset of Rkfor some k1.
Let hbe an arbitrary mesh on ¯
, and VhH1
0(Ω) a finite element space defined on this mesh.
Choose γRto satisfy 0< γ b0. Let ube the weak solution of (8) and uhthe solution of (9).
Then one has
|||uuh|||βC2
C1
inf
whVh|||uwh|||β.
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Proof. Let whVhbe arbitrary. Invoking Lemma 3, then equation (10), then Lemma 4, we get
C1|||uuh|||2
βBβ(uuh, u uh)
=Bβ(uuh, u wh)
C2|||uuh|||β|||uwh|||β.
Hence |||uuh|||β(C2/C1)|||uwh|||β. As whVhwas 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 |||uuh|||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(¯
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 uC2(¯
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)|0x1},Γ2:= {(0, y)|0y1},
Γ3:= {(x, 1)|0x1},Γ4:= {(1, y)|0y1}.
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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corner ck. There exists a constant Csuch that for all (x, y)¯
and 0m+n2one has
kDm
xDn
yvk0,,¯
C, (12a)
Dm
xDn
yw1(x, y)neb0y ,(12b)
Dm
xDn
yz1(x, y)mneb0(x+y) .(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=σεb1
0ln N. (14)
Partition Ω as follows (see Figure 1): ¯
Ω=Ω11 21 12 22, where
11 = [λ, 1λ]×[λ, 1λ],21 = ([0, λ][1 λ, 1]) ×[λ, 1λ],
12 = [λ, 1λ]×([0, λ][1 λ, 1]),
22 = ([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, 1λ] into N/2 equidistant subintervals. This gives a coarse mesh on [λ, 1λ] 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:= xixi1and kj:= yjyj1that are defined
by
hi=(h:= 4λN1for i= 0, . . . , N/4 and i= 3N/4+1, . . . , N,
H:= 2(1 2λ)N1for i=N/4+1,...,3N/4,
(15a)
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r r
r
r
λ
1λ
λ1λ
22
22
22
22
12
12
21 21
11
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 12, and fine on Ω22 .
The mesh is quasiuniform on Ω11 and its diameter dthere satisfies 2/N d22/N; on
12 21, each mesh rectangle has dimensions O(N1) by O(εN1ln N); and on Ω22 each
rectangle is O(εN1ln N) by O(εN1ln 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 VhC(¯
Ω) H1
0(Ω) to comprise
piecewise bilinears on the Shishkin mesh of Section 4.1. Given any function gC(¯
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 φH2(R). Then its bilinear nodal interpolant φIsatisfies the bounds
kφφIk,R Ch2
xkφxxk,R +k2
ykφyy k,R,
k(φφI)xk,R C(hxkφxxk,R +kykφxy k,R),
k(φφI)yk,R C(hxkφxyk,R +kykφyyk,R ),
where the constant Cis independent of φ, hxand ky.
Theorem 5 gives us
|||uuh|||βC2
C1|||uuI|||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
|||vvI|||βCεN1+N2.
Proof. Lemmas 6 and 8 yield kvvIk,CN2|v|2,,CN 2. Hence
kvvIkβ=Z
β(vvI)21/2
CN2Z
β1/2
CN2.
By a similar argument, again using Lemmas 6 and 8, one has k∇(vvI)k,CN1|v|2,,
CN1and hence k∇(vvI)kβC N 1. To finish the proof, recall the definition of ||| ·|||β.
For any measurable ωΩ, and each vL2(ω) and wH1(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
|||wjwI
j|||βCN1ln N+Nσfor j= 1,2,3,4.
Proof. We give the proof only for w1, as the other wjare similar.
Set Ωy={(x, y)Ω : yλ}. By Lemma 6, one has
Dm
xDn
yw1(x, y)nNσfor 0 m+n2 and (x, y)y.
Hence,
kw1wI
1k,y≤ kw1k,y+kwI
1k,y= 2kw1k,yCNσ
and
k∇(w1wI
1)k,y≤ k∇w1k,y+k∇wI
1k,y2k∇w1k,yCε1Nσ.
It now follows, as in the proof of Lemma 9, that
|||w1wI
1|||β,yCNσ.(17)
On \yinvoke Lemma 8 and (15) to get
kw1wI
1k,\yCH2k(w1)xxk,\y+h2k(w1)yyk,\y
CN2+ (εN1ln N)2ε2
=C(N1ln N)2
and
k∇(w1wI
1)k,\yCHk(w1)xxk,\y+ (h+H)k(w1)xy k,\y
+hk(w1)yy k,\y
CN1(1 + ε1)+(εN1ln N)(ε1+ε2)
=Cε1(N1ln 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
|||w1wI
1|||β,\yCN1ln 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
|||zjzI
j|||βCN1ln N+Nσfor j= 1,2,3,4.
Proof. We give the proof only for z1, as the other zjare similar.
Set Ωxy ={(x, y)Ω : xλor yλ}. By Lemma 6, one has
Dm
xDn
yw1(x, y)mnNσfor 0 m+n2 and (x, y)xy.
Hence,
kz1zI
1k,y≤ kz1k,y+kzI
1k,y= 2kz1k,yCNσ
and
k∇(z1zI
1)k,y≤ k∇z1k,y+k∇zI
1k,y2k∇z1k,yCε1Nσ.
It now follows, as in the proof of Lemma 9, that
|||z1zI
1|||β,yCNσ.(19)
On \xy invoke Lemma 8 and (15) to get
kz1zI
1k,\yCh2k(z1)xxk,\y+h2k(z1)yyk,\y
C(εN1ln N)2ε2
=C(N1ln N)2
and
k∇(z1zI
1)k,\yChk(z1)xxk,\y+hk(z1)xy k,\y
+hk(z1)yy k,\y
C(εN1ln N)(1 + ε1+ε2)
=Cε1(N1ln 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
|||z1zI
1|||β,\yCN1ln 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
|||uuh|||βCN1ln N+Nσ.
With the choice σ= 1 in the definition (14) of the Shishkin mesh transition parameter, this
bound becomes
|||uuh|||βCN1ln N.
Proof. Lemmas 9, 10 and 11 and the decomposition of Lemma 6 yield
|||uuI|||βCN1ln N+Nσ.
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
εv+avx+bv =fon Ω = (0,1)2, v = 0 on ,(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 , 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− |y1/2|]
ε1/2for (x, y).
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).
11
Preprint 20 November 2020
Table 1: |||uuh|||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
1012.975e-02 1.787e-02 1.043e-02 5.320e-03 2.660e-03
1023.322e-02 1.985e-02 1.157e-02 6.610e-03 3.718e-03
1033.360e-02 2.007e-02 1.169e-02 6.675e-03 3.754e-03
1043.358e-02 2.008e-02 1.170e-02 6.682e-03 3.758e-03
1053.366e-02 2.009e-02 1.170e-02 6.682e-03 3.758e-03
1063.366e-02 2.010e-02 1.170e-02 6.683e-03 3.758e-03
1073.366e-02 2.010e-02 1.170e-02 6.683e-03 3.758e-03
1083.366e-02 2.010e-02 1.170e-02 6.683e-03 3.758e-03
5 Numerical results
Our test problem is
ε2u+u=fin Ω = (0,1)2, u = 0 on ,(23)
where fchosen so that
u(x, y) = cos π
2xex/ε e1
1e1!1yey/ε e1
121.(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(N1ln 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 ε1/2N1ln 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(N1). We
remark that choosing σ= 2 in (13) improves the experimental convergence rate to O(N2ln2N),
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 |||uuh|||βstated
12
Preprint 20 November 2020
Table 2: |||uuh|||,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
1012.678e-03 9.603e-04 3.283e-04 8.664e-05 2.231e-05
1021.580e-02 4.857e-03 8.965e-04 1.549e-04 4.249e-05
1033.180e-02 1.445e-02 5.990e-03 2.090e-03 7.605e-04
1043.406e-02 1.670e-02 8.142e-03 3.912e-03 1.818e-03
1052.553e-02 1.398e-02 7.951e-03 4.132e-03 2.041e-03
1062.526e-02 1.306e-02 6.694e-03 3.413e-03 1.753e-03
1072.526e-02 1.306e-02 6.697e-03 3.412e-03 1.729e-03
1082.526e-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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14
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