Superconvergence analysis of FEM and SDFEM on

graded meshes for a problem with characteristic layers

M. Brdara,∗, G. Radojevb, H. -G. Roosc, Lj. Teofanovd

aFaculty of Technology, University of Novi Sad, Bulevar cara Lazara 1, 21000 Novi Sad,Serbia

bDepartment of Mathematics and Informatics, Faculty of Sciences, University of NoviSad, Trg Dositeja Obradovica 4, 21000 Novi Sad, Serbia

cInstitute of Numerical Mathematics, Technical University of Dresden, DresdenD-01062,Germany

dFaculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovica 6,21000 Novi Sad, Serbia

Abstract

We consider a singularly perturbed convection-diffusion with exponential andcharacteristic boundary layers. The problem is numerically solved by theFEM and SDFEM method with bilinear elements on a graded mesh. For theFEM we prove almost uniform convergence and superconvergence. The useof graded mesh allows for the SDFEM to prove almost uniform esimates inthe SD norm, which is not possible for Shishkin type meshes.

Keywords: singular perturbation, characteristic layers, finite elementmethod, streamline diffusion method, graded mesh, superconvergenceAMS Mathematics Subject Classification (2010): 65N12, 65N15, 65N30,65N50

∗Corresponding authorEmail addresses: mirjana.brdar@uns.ac.rs (M. Brdar), radojev@dmi.uns.ac.rs

(G. Radojev), hans-goerg.roos@tu-dresden.de (H. -G. Roos), ljiljap@uns.ac.rs(Lj. Teofanov)

Preprint submitted to Elsevier November 12, 2020

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1. Introduction

In this paper we consider the following convection-diffusion problem

Lu := −ε∆u− bux + cu = f in Ω = (0, 1)× (0, 1),

u = 0 on ∂Ω, (1)

with

b ∈ W 1,∞(Ω), c ∈ L∞(Ω), b ≥ β, c ≥ 0 on Ω,

where β is a positive constant and 0 < ε 1 a small perturbation parameter.We additionally assume that

c+1

2bx ≥ γ > 0, (x, y) ∈ Ω, (2)

for some constant γ, which will ensure the coercivity of the bilinear formassociated with the differential operator L.

Problem (1) belongs to the class of singularly perturbed problems whosesolutions are characterized by the so-called boundary layers - parts of thedomain where the solution changes abruptly, i.e. where the derivatives ofthe solution are very large. It is known that the presence of boundary layersin solutions of singularly perturbed problems makes the application of thestandard numerical procedures unstable and unsatisfactory. Therefore, theconstruction of at least almost ε-uniform numerical methods, which provideaccurate approximate solution in the whole domain, is the main issue in atreatment of boundary layer problems. Problem (1) can be used in modellingof flow past a surfice and the analysis of this problem and possibilities ofits numerical solutions can be of help in the numerical treatment of morecomplex problems.

The solution of problem (1) is characterized by an exponential layer atx = 0 and two parabolic layers at characteristic boundaries y = 0 and y = 1.Parabolic layers occur if the dominant part of the solution near some part ofthe boundary satisfies a partial differential equation of parabolic type. Thevariable in the streamline direction plays the role of the time variable in thisequation. This can be observed from the asymptotic expansion. It is knownthat for problems with parabolic layers does not exist a fitted scheme thatconverges uniformly on a uniform mesh [17]. Therefore, a general strategy

2

in the construction of a numerical method for the singularly perturbed prob-lem with parabolic layers is to apply some finite difference or finite elementmethod (FEM) on a specially designed layer-adapted mesh.

In this paper we will use a graded Duran-Lombardi (DL) mesh introducedin [2]. This mesh is defined implicitly by a recursive formula. It is a simpli-fied version of Gartland mesh from [8] where the recursive formula is basedon equidistribution of pointwise error, and contains exponential function init. DL mesh has the great advantage of being simple and not requiring thea priori definition of transition points. It is robust in the sense that a meshdefined for some fixed value of the perturbation parameter can also be usedfor lager values of the parameter. Moreover, for a certain range of ε this con-struction is even a better option then a mesh generated with a correspondingperturbation parameter. In the FE analysis done so far on DL meshes theirsimplicity was utilized to reduce initial assumptions on solution properties.

The streamline diffusion FEM (SDFEM) which adds weighted residualsto the Galerkin FEM is one of the most frequently studied and most popularstabilized FEM. This method was proposed first in [9] and applied to variousproblems. Compared with the standard Galerkin FEM, the SDFEM providesadditional control over the convective derivative in the streamline directionbecause of the definition of the induced streamline diffusion norm. Thisadditional bound prevents the discrete solution from oscillating over a largepart of the domain. It is well known that the SDFEM has high accuracyaway from layers and good stability properties. However, in layer regions theSDFEM fails to compute accurate solutions unless layer-adapted meshes areused. There are lots of results for singularly perturbed problems concerningSDFEM on Shishkin meshes; here we refer to some of them dealing withproblem (1): [5, 7, 12, 14, 19]. In the analysis of the SDFEM on Shishkintype meshes it is not possible to prove the desired estimates in the SD norm,instead only an related energy norm is used. This remedy was improvedby Zhang and Liu [13], they used modified SD norm. Franz [4] used thesame trick to modify the LPS norm for the local projection stabilizationmethod. Surprisingly, for graded meshes estimates in the original SD normare possible, see [18] for a problem with exponential layers only. We showthat this can also be realized for problems with exponential and characteristiclayers.

In this paper we give for the first time a detailed superconvergence anal-ysis of the SDFEM with bilinear elements on a DL mesh for problem (1).This analysis provides a certain choice of the streamline diffusion parameters.

3

The optimal choice of SD parameter on anisotropic meshes is still an openproblem. For one-dimensional problems, the SD parameter can be chosen insuch a way that the discrete solution is exact in mesh points or that it makesthe coefficient matrix an M-matrix. For a two-dimensional problem, this ap-proach can not be applied. Furthermore, for a problem with characteristicboundary layers it is more difficult to tune the SD parameter.

The paper is organized as follows. In Section 2 we set an assumptionregarding solution properties of the problem (1). In Section 3 we describelayer-adapted DL mesh and give some auxiliary estimates on the solutionderivatives in L2 norm. Sections 4 and 5 contain proofs of convergence andsuperconvergence of Galerkin FEM on a DL mesh. The formulation and basicfeatures of SDFEM are given in Section 6. Section 7 is devoted to the proofof superconvergence result of SDFEM on a DL mesh under a certain choiceof the SD parameter. Finally, numerical results are presented in Section 8,and a summary of the results is given in Section 9.

Notation 1. For a set D, a standard notation for Banach spaces Lp(D),Sobolev spaces W k,p(D), Hk(D) = W k,2(D), norms ‖ · ‖Lp(D) and seminorms| · |Hk(D) are used. Specially, if p = 2 we denote the norm with ‖ · ‖0,D.The standard scalar product in L2(D) is marked with (·, ·)D. Throughout thepaper, we often use notation A . B if a generic constant C independent ofε and mesh parameter h exists, such that A ≤ CB.

2. Solution properties

The forthcoming error analysis is based on some a priori knowledge aboutthe behaviour of the problem solution. Therefore, in the following assump-tion we give a solution decomposition and bounds of their components andderivatives. The validity of this assumption for constant coefficient problemunder sufficient smoothness and compatibility conditions for function f isproved in [10, 15]. Information about the location of layers obtained fromthese estimates is also important for the construction of a layer-adaptedmesh.

4

Assumption 2.1. [6] The solution u of problem (1) can be decomposed asu = v + w1 + w2 + w12, where for all x, y ∈ [0, 1] and 0 ≤ i+ j ≤ k we have∣∣∂ix∂jyv(x, y)

∣∣ . 1,∣∣∂ix∂jyw1(x, y)∣∣ . ε−i exp−βx/ε, (3)∣∣∂ix∂jyw2(x, y)∣∣ . ε−j/2(exp−yδ/

√ε + exp−(1−y)δ/

√ε),∣∣∂ix∂jyw12(x, y)

∣∣ . ε−(i+j/2) exp−βx/ε(exp−yδ/√ε + exp−(1−y)δ/

√ε)

for some δ > 0. For 0 ≤ i+ j ≤ k + 1 the L2 bounds are

||∂ix∂jyv(x, y)||0,Ω . 1, ||∂ix∂jyw1(x, y)||0,Ω . ε−i−1/2, (4)

||∂ix∂jyw2(x, y)||0,Ω . ε−j/2+1/4, ||∂ix∂jyw12(x, y)||0,Ω . ε−i−j/2+1/4.

The solution u of problem (1) satisfies∣∣∂ix∂jyu(x, y)∣∣ . (1 + ε−i exp−βx/ε +ε−j/2(exp−yδ/

√ε + exp−(1−y)δ/

√ε)

+ε−(i+j/2) exp−βx/ε(exp−yδ/√ε + exp−(1−y)δ/

√ε))

(5)

for all (x, y) ∈ Ω, 0 ≤ i+ j ≤ k, and some δ > 0.

Remark 1. Note that the estimate (5) follows from (3). In the classicalanalysis from [2, 3] on graded meshes the authors use only estimates forthe derivatives like (5) instead a solution decomposition. We prefer to use alittle bit stronger assumption for the solution decomposition to obtain a bettersuperconvergence result for SDFEM, see Section 7.

3. The graded DL mesh

The graded DL mesh was introduced in [2] and here we adapt it forproblem (1). For a given parameter 0 < h < 1, the mesh in x-direction Ωh

x isrecursively graded with mesh points given by

x0 = 0,

xi = ihε, 1 ≤ i ≤ d 1he

xi+1 = xi + hxi, d 1he ≤ i ≤Mx − 2,

xMx = 1,

(6)

5

and in y-direction we obtain Ωhy with

y0 = 0,

yj = jh√ε, 1 ≤ j ≤ d 1

he

yj+1 = yj + hyj, d 1he ≤ j ≤My − 2,

yMy = 1/2,

yMy+j = 1− yMy−j, j = 1, . . . ,My,

(7)

where integers Mx and My are such that the following inequalities are valid

xMx−1 < 1 and xMx−1(1 + h)≥ 1,

yMy−1 <1

2and yMy−1(1 + h)≥ 1

2.

We also assume that the last interval (xMx−1, 1) is not too small comparedto the previous one (xMx−2, xMx−1). If this is not the case the node xMx−1

should be eliminated. Analogously in y-direction. Then the DL mesh on Ωis given by the following tensor product

Ωh = Ωhx × Ωh

y . (8)

Set Nx =

⌊1

h+ 1

⌋. Then xNx = Nxhε,

xMx−1 = Nxhε(1 + h)Mx−Nx−1 < 1 Nxhε(1 + h)Mx−Nx ≥ 1

and

yMy−1 = Nxh√ε(1 + h)My−Nx−1 <

1

2Nxh√ε(1 + h)My−Nx ≥ 1

2.

From the above inequalities, we obtain

h

2ε(1 + h)Mx−Nx−1 <

1

2N−1x ≤ h

√ε(1 + h)My−Nx

and

h√ε(1 + h)My−Nx−1 <

1

2N−1x ≤

1

2hε(1 + h)Mx−Nx .

A simple calculation gives a relation between My and Mx, i.e.

ln√ε

2

ln (1 + h)+Mx − 1 < My <

ln√ε

2

ln (1 + h)+Mx + 1.

6

Figure 1: DL mesh for h = 0.5, ε = 10−3 (left) and h = 0.3, ε = 10−6 (right).

Let the lower and the upper bound for My be denoted by

L =

⌈ln√ε

2

ln (1 + h)

⌉+Mx − 1, U =

⌊ln√ε

2

ln (1 + h)

⌋+Mx + 1. (9)

Table 1 shows (9) for fixed ε and different values of h. The mesh sizeshx,i = xi − xi−1, i = 1, . . . ,Mx, have the following properties

hx,i = hε, 1 ≤ i ≤ d 1he

hx,i ≤ hx, x ∈ [xi−1, xi], d 1he+ 1 ≤ i ≤Mx.

(10)

The mesh sizes hy,i = yi − yi−1, i = 1, . . . , 2My satisfy

hy,j = h√ε, 1 ≤ j ≤ d 1

he, 2My − d 1

he ≤ j ≤ 2My

hy,j ≤ hy, y ∈ [yj−1, yj], d 1he+ 1 ≤ j ≤ 2My − d 1

he − 1.

(11)

For the most layer-adapted meshes constructed based on a priori giveninformation, the number of mesh points is given in advance. For a DL meshthere is no unique parameter h that generates a mesh with a fixed numberof mesh nodes N . To compare numerical results on a DL mesh with resultsobtained on other meshes, we construct a DL mesh which gives unique h,

7

h Mx My L U0.100 107 52 51 520.095 113 54 54 550.090 136 57 56 570.085 125 60 60 610.080 133 64 64 650.075 141 68 67 680.070 151 73 72 730.065 162 78 77 780.060 175 84 84 850.055 191 92 92 930.050 209 125 124 125

Table 1: Lower and upper bounds L and U for ε = 10−4

for a given N . One additional condition should be imposed in each directionto obtain this property. For a fixed N , the parameters hx and hy will becalculated such that in (6) and (7) we have

Mx = N, xN = xN−1 + hxxN−1 = 1 (12)

My =N

2, yN/2 = yN/2−1 + hyyN/2−1 =

1

2(13)

respectively.Since the widths of characteristic and exponential boundary layers are

O(√ε| ln ε|) and O(ε| ln ε|) respectively ([11],[16, p.274]), the domain Ω is

consequently divided into the following subdomains:

Ω1 =⋃Rij : xi−1 < c1ε| ln ε|,

Ω2 =⋃Rij : xi−1 ≥ c1ε| ln ε|, yj−1 < c2

√ε| ln ε| or yj−1 > 1− c2

√ε| ln ε|,

Ω3 =⋃Rij : xi−1 ≥ c1ε| ln ε|, c2

√ε| ln ε| ≤ yj−1 ≤ 1− c2

√ε| ln ε|,

where Rij = [xi, xi−1]× [yj, yj−1] and constants c1 and c2 are such that∣∣∣ ∂i+ju∂xi∂yj

∣∣∣ ≤ C for 0 ≤ i+ j ≤ 3,

if x > c1ε| ln ε| and c2

√ε| ln ε| < y < 1− c2

√ε| ln ε|.

8

Ω1 Ω3

Ω2

Ω2

Figure 2: Partitioning of the domain Ω

In view of Assumption 2.1, it is enough to take

c1 >3

βand c2 >

3

2δ. (14)

Lemma 1. [2] If 0 < h < 1 is a mesh parameter and Mx is the number ofmesh points in Ωh

x, thenh .M−1

x | ln ε|.

Analogously, h .M−1y | ln

√ε|. Therefore, if the total number of mesh points

is denoted by M , then

h .1√M| ln ε|. (15)

In the following two lemmas several a priori estimates for the solutionof problem (1) are given. They will be frequently used in convergence andsuperconvergence analysis.

Lemma 2. The solution u of problem (1) satisfy

||uxx||20,Ω . ε−3, ||uyy||20,Ω . ε−3/2, ||xuxx||20,Ω . ε−1, ||x2uxx||20,Ω . 1,

||yuxy||20,Ω . ε−1, ||uxy||20,Ω . ε−3/2, ||xyuxy||20,Ω . 1, ||xuxy||20,Ω . ε−1/2,

||y2uyy||20,Ω . 1, ||yuyy||20,Ω . ε−1/2.

9

Lemma 3. For the solution u of problem (1), we have the following a prioriestimates

‖x3uxxx‖0,Ω . 1, ‖y3uyyy‖0,Ω . 1, ‖x2uxxx‖0,Ω . ε−1/2,

‖y2uyyy‖0,Ω . ε−1/4, ‖xuxxx‖0,Ω . ε−3/2, ‖yuyyy‖0,Ω . ε−3/4,

‖uxxx‖0,Ω . ε−5/2, ‖uyyy‖0,Ω . ε−5/4, ‖uxxy‖0,Ω . ε−7/4,

‖uxyy‖0,Ω . ε−5/4, ‖x2uxxy‖0,Ω . ε−1/4, ‖y2uxyy‖0,Ω . ε−1/2.

Moreover, if (14) is satisfied then

‖uxxx‖0,Ω1 . ε−5/2, ‖yuxxy‖0,Ω1 . ε−3/2, ‖y2uxyy‖0,Ω1 . ε−1/2,

‖x2uxxx‖0,Ω2 . ε1/4| ln ε|1/2, ‖xuxxy‖0,Ω2 . ε−1/4, ‖uxyy‖0,Ω2 . ε−3/4,

‖x2uxxx‖0,Ω3 . 1, ‖y2uyyy‖0,Ω3 . 1, ‖x2uxxy‖0,Ω3 . 1,

‖y2uxyy‖0,Ω3 . 1, ‖xyuxxy‖0,Ω3 . 1, ‖xyuxyy‖0,Ω3 . 1,

‖xuxxx‖0,Ω3 . 1, ‖yuyyy‖0,Ω3 . 1, ‖xuxyy‖0,Ω3 . 1,

‖yuxxy‖0,Ω3 . 1.

Proofs of Lemmas 2 and 3 are based on Assumption 2.1 and mesh size prop-erties (10) and (11). They are analogous to the proofs of related estimatesin [2].

4. FEM

The main goal of this paper is a superconvergence result for SDFEM on aDL mesh. Therefore, the following convergence and superconvergence resultsfor FEM on a DL mesh are necessary ingredients. The corresponding proofsare mainly based on techniques from [2, 3] and we present details just whenthey differ because of the nature of problem (1).

For problem (1), the standard weak formulation is:find u ∈ H1

0 (Ω) such that aG(u, v) = (f, v), ∀v ∈ H10 (Ω), with the bilinear

form

aG(w, v) := ε1(∇w,∇v)− (bwx + cw, v), w, v ∈ H10 (Ω).

Let V h ⊂ H10 (Ω) be the finite element space of piecewise bilinear functions

defined on DL mesh (8). The Galerkin finite element method is characterized

10

by: find uh ∈ V h such that

aG(uh, vh) = (f, vh), ∀vh ∈ V h. (16)

The bilinear form satisfies the Galerkin orthogonality property. The bilinearform aG(·, ·) is coercive with respect to the energy norm

‖u‖2ε := ε|u|21 + ‖u‖2

0 (17)

due to assumption (2). Hence, the standard weak formulation and theGalerkin method have unique solutions.

Given any u ∈ C0(Ω) and a triangulation TN of Ω into rectangles wedenote by uI the nodal piecewise bilinear interpolant to u over TN . Let uh

be the FEM solution.

Theorem 1. Let u be the solution of (1). If (5) with k = 2 holds true, thenon DL mesh (8) the interpolation error satisfies

‖u− uI‖0,Ω . h2, ‖u− uI‖ε . h.

Proof: We use the estimate from [1, Theorem 3] and estimates given inLemma 2 to obtain

d 1he∑

i=1

d 1he∑

j=1

‖u− uI‖20,Rij. h4ε1/2,

d 1he∑

i=1

My−d 1h e∑j=d 1

he+1

‖u− uI‖20,Rij. h4,

Mx∑i=d 1

he+1

d 1he∑

j=1

‖u− uI‖20,Rij. h4,

Mx∑i=d 1

he+1

My−d 1h e∑j=d 1

he

‖u− uI‖20,Rij. h4,

which imply ‖u − uI‖0,Ω . h2. Similarly, we get ‖(u − uI)x‖20,Ω . h2ε−1,

specifically

‖(u− uI)x‖20,Ω1. h2ε−1, ‖(u− uI)x‖2

0,Ω2. h2

√ε| ln ε|, ‖(u− uI)x‖2

0,Ω3. h2,

(18)

and ‖(u− uI)y‖20,Ω . h2ε−1/2, so the theorem holds true.

Theorem 2. Let u be the solution of (1). If (5) with k = 2 holds true, thenfor the approximate solution uh obtained by FEM with bilinear elements onDL mesh (8) we have

‖u− uh‖ε . h| ln ε| . 1√M

ln2 ε.

11

Proof: Let η = u − uI , χ = uI − uh. From Galerkin orthogonality we haveaG(u − uh, χ) = aG(η, χ) + aG(χ, χ). The bilinear form aG(·, ·) is coercivewith respect to the energy norm (17). Let α = min1, γ, then coercivityand Galerkin orthogonality imply

α‖χ‖2ε ≤ aG(χ, χ) = −aG(η, χ) ≤

∣∣∣− ε(∇η,∇χ)− (bη, χx)− ((c+ bx)η, χ)∣∣∣

≤ C(‖u− uI‖ε‖χ‖ε +

∫Ω

b(u− uI)xχdΩ)

≤ C(‖u− uI‖2

ε +α

2‖χ‖2

ε +

∫Ω

b(u− uI)xχdΩ).

Then using Theorem 1 we have

α

2‖χ‖2

ε ≤ Ch2 + C

∫Ω

b(u− uI)xχdΩ (19)

On Ω1 Poincare inequality gives ‖χ‖0,Ω1 ≤ Cε| ln ε|‖∇χ‖0,Ω1 , using the es-timates from the proof of Theorem 1, and generalized arithmetic-geometricmean inequality we obtain∫Ω1

b(u− uI)xχdΩ1 ≤ C‖(u− uI)x‖0,Ω1‖χ‖0,Ω1

≤ C‖(u− uI)x‖0,Ω1ε| ln ε|‖∇χ‖0,Ω1 ≤ Cε ln2 ε‖(u− uI)x‖20,Ω1

+ ρε‖∇χ‖20,Ω1

≤ Ch2 ln2 ε+ ρε‖∇χ‖20,Ω1

. (20)

Using the same arguments on Ω2 we get∫Ω2

b(u− uI)xχdΩ2 ≤ Ch2 ln2 ε+ ρ‖∇χ‖20,Ω2

. (21)

Also, on Ω3 we have∫Ω3

b(u− uI)xχdΩ3 ≤ C‖(u− uI)x‖0,Ω3‖χ‖0,Ω3 ≤ C‖(u− uI)x‖20,Ω3

+ ρ‖χ‖20,Ω3

≤ Ch2 + ρ‖χ‖20,Ω3

. (22)

12

For ρ small enough from (20)-(22) we obtain∫Ω

b(u − uI)xχdΩ . h2 ln2 ε.

Therefore, (19) implies ‖χ‖ε ≤ Ch| ln ε| which together with Theorem 1completes the proof.

One should remark that Theorem 2 also holds for linear elements.

5. Superconvergence of the FEM

In this section, we prove that the finite element approximation definedabove has a property that the difference between the computed solution andthe Lagrange interpolant of the exact solution is of higher order than the erroritself. The forthcoming lemmas provide necessary estimates that contributeto the proof of superconvergence result.

Lemma 4. The solution uh of Galerkin discretization (16) satisfies∣∣∣ε ∫Ω

∇(u− uI)∇χdxdy∣∣∣ . h2‖χ‖ε for all χ ∈ V h.

Proof: By Lemma 3 and approach used in the proof of Theorem 1, fourestimates are obtained:

ε

d 1he∑

i=1

d 1he∑

j=1

‖(u− uI)xvx‖20,Rij. ε(h2

x,i‖uxxx‖0,Rij+ hx,ihy,j‖uxxy‖0,Rij

+ h2y,j‖uxyy‖0,Rij

)‖vx‖0,Rij. ε(h2ε2‖uxxx‖0,Rij

+ h2ε√ε‖uxxy‖0,Rij

+ h2ε‖uxyy‖0,Rij)‖vx‖0,Rij

. ε1/4h2‖v‖ε,

ε

d 1he∑

i=1

My−d 1h e∑j=d 1

he+1

‖(u− uI)xvx‖20,Rij. ε(h2ε2‖uxxx‖0,Rij

+ h2ε‖yuxxy‖0,Rij

+ h2‖y2uxyy‖0,Rij)‖vx‖0,Rij

. h2‖v‖ε,

εMx∑

i=d 1he+1

d 1he∑

j=1

‖(u− uI)xvx‖20,Rij. ε(h2‖x2uxxx‖0,Rij

+ h2√ε‖xuxxy‖0,Rij

+ h2ε‖uxyy‖0,Rij)‖vx‖0,Rij

. ε1/4h2‖v‖ε,

13

ε

Mx∑i=d 1

he+1

My−d 1h e∑j=d 1

he

‖(u− uI)xvx‖20,Rij. ε(h2‖x2uxxx‖0,Rij

+ h2‖xyuxxy‖0,Rij

+ h2‖y2uxyy‖0,Rij)‖vx‖0,Rij

. h2‖v‖ε.

Analogously, similar estimates for ε‖(u − uI)yvy‖20,Ω are obtained, so the

lemma follows.

Lemma 5. The solution uh of Galerkin discretization (16) satisfies∣∣∣ ∫Ω

b(u− uI)xχdxdy∣∣∣ . h2| ln ε|

12‖χ‖ε for all χ ∈ V h.

Proof: Here we use technique from [6]. In Assumption 2.1 the solutiondecomposition u = v + w1 + w2 + w12 is introduced. Let w = w1 + w12, and

Ω0 =⋃Rij : xi−1 < c1ε| ln ε|, yi−1 < c2

√ε| ln ε| ⊂ Ω1.

Then integration by parts yields

(b(u− uI)x, χ) = (b(v − vI)x, χ) + (b(w2 − wI2)x, χ)Ω0∪Ω2 − (bx(w − wI), χ)

−(b(w − wI), χx)− (bx(w2 − wI2), χ)(Ω1\Ω0)∪Ω3 − (b(w2 − wI2), χx)(Ω1\Ω0)∪Ω3 ,

since χ ∈ V h vanishes on the boundary of Ω. For the terms on the right-handside we have the following estimates,

|(bx(w − wI), χ)|+ |(bx(w2 − wI2), χ)(Ω1\Ω0)∪Ω3| . h2‖χ‖ε|(b(w2 − wI2), χx)(Ω1\Ω0)∪Ω3|+ |(b(w − wI), χx)| . h2‖χ‖ε

|(b(v − vI)x, χ)| . h2| ln ε|12‖χ‖ε

|(b(w2 − wI2)x, χ)Ω0∪Ω2| . h2| ln ε|12‖χ‖ε.

Lemma 6. The solution uh of Galerkin discretization (16) satisfies∣∣∣ ∫Ω

c(u− uI)χdxdy∣∣∣ . h2‖χ‖ε for all χ ∈ V h.

14

The proof follows from Cauchy-Schwarz inequality and Theorem 1.

Theorem 3. Let Assumption 2.1 holds true. Then the FEM solution uh

obtained on DL mesh (8) with bilinear elements and uI ∈ V h satisfy

‖uI − uh‖ε . h2| ln ε|12 .

1

M| ln ε|

52 . (23)

Proof: The analysis starts from

‖uI − uh‖2ε ≤ |aG(η, χ)| ≤ ε|(∇η,∇χ)|+ |(bηx, χ)|+ |c(η, χ)|. (24)

Above, in Lemmas 4-6, we give estimates for each of the right-hand sideterms. Finally, dividing (24) by ‖uI−uh‖ε the statement of theorem follows.

Remark 2. If we do not use the solution decomposition, but only (5) withk = 3 and the technique from [3] we obtain∣∣∣ ∫

Ω

b(u− uI)xχdxdy∣∣∣ . h2| ln ε|3‖χ‖ε for all χ ∈ V h,

so the superconvergence result then is

‖uI − uh‖ε ≤ Ch2| ln ε|3 . 1

M| ln ε|5. (25)

The superconvergence result can be further used to improve numericalapproximation by using some postprocessing approach.

6. SDFEM

In order to stabilize the discretization given by the standard GalerkinFEM we introduce the streamline diffusion FEM

aG(w, v) +∑τ∈ΩN

%τ (f − Lw, bvx)τ = (f, v),

where %τ ≥ 0 is a user chosen parameter. Its discretization reads: Finduh ∈ V h such that

aSD(uh, vh) := aG(uh, vh) + astab(uh, vh) = fSD(vh) for all vh ∈ V h, (26)

15

withastab(w, v) :=

∑τ∈ΩN

%τ (ε∆w + bwx − cw, bvx)τ ,

fSD(v) := (f, v)−∑τ∈ΩN

%τ (f, bvx)τ .

This bilinear form also satisfies the orthogonality condition. Now we definea streamline diffusion norm

‖v‖2SD := ‖v‖2

ε +∑τ∈ΩN

%τ‖bvx‖20,τ .

It is shown in [16] that if

0 ≤ %τ ≤ γ/‖c‖20,τ , τ ∈ TN , (27)

then aSD(v, v) ≥ 12‖v‖2

SD. Note that ‖v‖ε ≤ ‖v‖SD for all v ∈ H10 (Ω). There-

fore, aSD(·, ·) has a stronger stability then aG(·, ·). Roughly, the method ismore stable when %τ is closer to its upper bound. Problem (26) has a uniquesolution uh ∈ V h.

We propose the following choice for the SD parameter:

%1 . εM−1, %2 . ε−14M−1, %3 .

ε−1M−1, M− 1

2 ≤ ε

M− 12 , M− 1

2 ≥ ε(28)

on Ω1,Ω2 and Ω3 respectively.One should remark that these bounds for the stabilization parameters are

in accordance with the result of [7]. Moreover, on a DL mesh it is possibleto prove interpolation error estimate in the SD norm.

Theorem 4. Let u be the solution of (1). If (5) with k = 2 holds true, thenon DL mesh (8) the interpolation error in SD norm with (28) satisfies

‖u− uI‖SD . h .1√M| ln ε|.

The proof of Theorem 4 follows directly from Theorem 1, estimates (18), and(28).

16

7. Superconvergence of the SDFEM

In this section, we prove that the SDFEM approximation defined abovealso has a superconvergence property. In the proof of Theorem 5 the followinglemma will be frequently used.

Lemma 7. [7] Let b ∈ W 1,∞(Ω). Then∣∣∣(b(ϕ− ϕI)x, bχx)Ωi

∣∣∣[(hx,i + hy,i)(hx,i‖ϕxx‖0,Ωi

+ hy,i‖ϕxy‖0,Ωi) + h2

y,i‖ϕxyy‖0,Ωi

]‖χx‖0,Ωi

for i = 1, 2, 3.

Theorem 5. Let Assumption 2.1 holds true. Suppose the stabilization pa-rameter satisfies (27) and (28). Then the streamline diffusion approximationuh obtained on DL mesh with bilinear elements and uI ∈ V h satisfy

‖uI − uh‖SD .1

M| ln ε|

52 . (29)

Proof: In our error analysis we start from the coercivity and Galerkin or-thogonality:

1

2‖χ‖2

SD ≤ aG(η, χ) + astab(η, χ). (30)

From the proof of Theorem 3 we have |aG(η, χ)| . h2| ln ε| 12‖χ‖ε, so thesecond term in (30) has to be estimated

astab(u− uI , χ) =∑τ∈TN

%τ

(ε(∆(u− uI), bχx)

+ (b(u− uI)x, bχx)− (c(u− uI), bχx)). (31)

We estimate these tree terms separately on different subdomains of Ω. Esti-mates from Assumption 2.1 are frequently used in the following. For the thirdterm in (31) we use Cauchy-Schwarz inequality and Theorem 1 to obtain

%2|(c(u− uI), bχx)Ω2| ≤ C%2‖u− uI‖0,Ω2‖bχx‖0,Ω2 ≤ C%2h2ε

14 | ln ε|

12‖bχx‖0,Ω2

≤ C%122 h

2ε14 | ln ε|

12‖χ‖SD (32)

17

For the second term in (31) let w = w1 + w12 and w = v + w2.

%2|(b(w − wI)x, bχx)Ω2 | ≤ C%2

(‖wx‖L1(Ω2)‖bχx‖L∞(Ω2) + ‖wIx‖0,Ω2‖bχx‖0,Ω2

)≤ C%2

(‖wx‖L1(Ω2)ε

− 14 | ln ε|−

12‖bχx‖0,Ω2 + ε

14 | ln ε|

12‖wIx‖L∞(Ω2)‖bχx‖0,Ω2

)≤ C%2

(εc1β+ 1

4 | ln ε|12‖bχx‖0,Ω2 + εc1β−

34 | ln ε|

12‖bχx‖0,Ω2

)≤ C%

122 ε

c1β− 34 | ln ε|

12‖χ‖SD ≤ C%

1/22 ε

94 | ln ε|

12‖χ‖SD (33)

based on (14). If we use Lema 7, we get

%2|(b(w − wI)x, bχx)Ω2|

≤ C%2

((h+ h)(h‖xwxx‖0,Ω2 + h

√ε‖wxy‖0,Ω21 + h‖ywxy‖0,Ω22)

+ h2ε‖wxyy‖0,Ω21 + h2‖y2wxyy‖0,Ω22

)‖bχx‖0,Ω2

≤ C%122 ε

14h2‖χ‖SD, (34)

where Ω21 is a part of Ω2 where hy,i is equidistant while Ω22 is a part ofΩ2 where hy,i is non-equidistant. For the first part in (31), using Holderinequality, we get

ε%2|(∆w, bχx)Ω2| ≤ Cε%2‖∆w‖L1(Ω2)‖bχx‖L∞(Ω2)

≤ Cε%2εc1β− 1

2 | ln ε|‖bχx‖L∞(Ω2)

≤ Cε%2εc1β− 1

2 | ln ε|ε−14 | ln ε|−

12‖bχx‖0,Ω2

≤ C%122 ε

c1β− 34 | ln ε|

12‖χ‖SD. (35)

From(∆w, bχx)Ω2 + (∆w, bχx)Ω0 = −((b∆w)x, χ)Ω2∪Ω0 ,

where Ω0 = (0, ε| ln ε|)× (0,√ε| ln ε|), we have

ε%2|(∆w, bχx)Ω2| ≤ Cε%2

(‖(∆w)x‖0,Ω2∪Ω0‖χ‖0,Ω2∪Ω0 + ‖∆w‖0,Ω0‖χx‖0,Ω0

)≤ Cε%2(ε−

34‖χ‖0,Ω2∪Ω0 + ε−

14 | ln ε|‖χx‖0,Ω0)

≤ C%2ε14 | ln ε|‖χ‖ε. (36)

18

Collecting the above results (32)-(36), we get

astab(u− uI , χ)Ω2 ≤ C(%2ε

14 | ln ε|+ %

122 h

2ε14 | ln ε|

12

)‖χ‖SD. (37)

For the third term in (31) on Ω1 we obtain

%1|(c(u− uI), bχx)Ω1| ≤ C%1‖u− uI‖0,Ω1‖bχx‖0,Ω1 ≤ C%121 h

2‖χ‖SD (38)

For the second term we proceed as follows. Let w = w2 +w12 and w = v+w1.

%1|(b(w − wI)x, bχx)Ω1 | ≤ C%1

(‖wx‖L1(Ω1)‖bχx‖L∞(Ω1) + ‖wIx‖0,Ω1‖bχx‖0,Ω1

)≤ C%

121 ε− 1

2 | ln ε|12‖χ‖SD. (39)

Using Lemma 7, the following holds

%1|(b(w − wI)x, bχx)Ω1|

≤ C%1

(2h(hε‖wxx‖0,Ω11 + h‖xwxx‖0,Ω12 + h

√ε‖wxy‖0,Ω21 + h‖ywxy‖0,Ω22)

+ h2ε‖wxyy‖0,Ω21 + h2‖y2wxyy‖0,Ω22

)‖bχx‖0,Ω1

≤ C%121 ε− 1

2h2‖χ‖SD, (40)

where Ω21 is a part of Ω1 where hy,i is equidistant while Ω22 is a part of Ω1

where hy,i is non-equidistant. Analogously, Ω11 is a part of Ω1 where hx,i isequidistant while Ω12 is a part of Ω1 where hx,i is non-equidistant. For thefirst part in (31) we get

ε%1|(∆w, bχx)Ω1| ≤ Cε%1‖∆w‖L1(Ω1)‖bχx‖L∞(Ω1) ≤ C%121 | ln ε|

12‖χ‖SD (41)

and

ε%1|(∆w, bχx)Ω1| ≤ Cε%1‖∆w‖0,Ω1‖χx‖0,Ω1 ≤ C%1ε−1‖χ‖ε. (42)

From (38)-(42), we get

astab(u− uI , χ)Ω1 ≤ C(%1ε−1 + %

121 ε− 1

2h2| ln ε|12

)‖χ‖SD. (43)

19

On Ω3 let w = w1 + w2 + w12. Then Cauchy-Schwarz inequality implies

ε%3|(∆w, bχx)Ω3| ≤ Cε%3‖∆w‖0,Ω3‖bχx‖0,Ω3 ≤ C%3ε54‖χ‖ε (44)

and

%3|(c(w − wI), bχx)Ω3| ≤ C%3‖w − wI‖0,Ω3‖bχx‖0,Ω3 ≤ C%123 h

2‖χ‖SD (45)

The following identity from [6]

|((v − vI)x, χx)τ | ≤ Ch2y,τ‖vxyy‖0,τ‖χx‖0,τ for all v ∈ C3(τ) (46)

applied to the second term gives

%3|(b(w − wI)x, bχx)Ω3| ≤ C%3h2‖y2wxyy‖0,Ω3‖χx‖0,Ω3

≤ C%123 ε

74 | ln ε|2h2‖χ‖SD ≤ C%

123 h

2‖χ‖SD. (47)

The terms containing v are not exponentially small away from the layers,so they require more careful treatment. We use

(∆v, bχx)Ω3 + (∆v, bχx)Ω1\Ω0 = −((b∆v)x, χ)Ω3∪(Ω1\Ω0)

and obtain

ε%3|(∆v, bχx)Ω3| ≤ Cε%3(‖χ‖0,Ω1∪Ω3 + ‖χx‖L1(Ω1\Ω0))

≤ Cε%3(‖χ‖ε + (meas(Ω1))12‖χx‖0,Ω1\Ω0)

≤ C%3ε| ln ε|12‖χ‖ε. (48)

Here we also use (46)

%3|(b(v − vI)x, bχx)Ω3| ≤ C%3h2‖bχx‖0,Ω3 ≤ C%

123 h

2‖χ‖SD. (49)

%3|(c(v − vI), bχx)Ω3| ≤ C%3‖v − vI‖0,Ω3‖bχx‖0,Ω3 ≤ C%123 h

2‖χ‖SD (50)

Collecting (44)-(50), we obtain

astab(u− uI , χ)Ω3 ≤ C(%3ε| ln ε|

12 + %

123 h

2)‖χ‖SD. (51)

Applying estimates (28) of the SD parameters in (37), (43), and (51),together with (15), we obtain (29).

20

Figure 3: Test problem with ε = 10−3.

8. Numerical experiments

In this section, we present some numerical experiments in order to testour theoretical results. We consider model problem from [7] given by

−ε∆u− (2− x)ux +3

2u = f(x, y) x ∈ Ω,

u = 0, on ∂Ω,

where the function f is chosen in such a way that

u(x, y) =(

cosπx

2− e−x/ε − e−1/ε

1− e−1/ε

)(1− e−y/√ε)(1− e−(1−y)/

√ε)

1− e−1/√ε

is the exact solution. The rate of convergence is calculated in the standardway. All computations were carried out using MATLAB R2020a. We employSDFEM with ρτ chosen to be the maximal value allowed by Theorem 5.

The errors in various norms for SDFEM are presented in Tables 2-4. Ta-ble 2 shows the results on a DL-mesh when parameter h is chosen a priori.

21

The superconvergence property can be very well observed. In Table 3 pa-rameter ε is fixed while the number of mesh points in both directions is givenin advance. Here we can clearly see the first order of convergence and thesecond order of superconvergence result.

In Table 4 the number of mesh points in each direction is fixed until εvaries. This table shows that SDFEM on a DL-mesh is almost uniform inthe singular perturbation parameter ε (up to the logarithmic factor).

The last column in Tables 2-4 contain the errors in L∞ norm which sug-gest second order convergence. These results are given for the purpose ofcomparison with finite difference methods. We do not have theoretical justi-fication for it.

Finally, in Table 5 the comparison between superconvergence property ofSDFEM method ona a DL mesh and a Shishkin mesh for fixed ε is given.Numerical experiments shows that SDFEM method gives better results ona DL mesh than on a Shishkin mesh for the number of mesh points greaterthen 322.

h Mx 2My ‖u− uh‖SD ‖uI − uh‖SD ‖u− uh‖∞0.15 106 104 3.312e-02 1.728e-03 2.693e-030.09 172 168 1.997e-02 6.117e-04 1.077e-030.06 254 248 1.335e-02 2.695e-04 5.065e-040.04 378 368 8.914e-03 1.183e-04 2.334e-040.02 748 728 4.465e-03 2.971e-05 6.066e-050.085 1750 1704 1.900e-03 5.328e-06 1.113e-05

Table 2: Errors on mesh DL (6)-(7) with ε = 10−6 and different h.

9. Summary

A singularly perturbed elliptic problem with characteristic layer has beenconsidered. To obtain numerical approximation of the problem, we applyGalerkin FEM and SDFEM with bilinear elements on a layer-adapted DLmesh. We proved that such discretizations exhibit superconvergence propertywith the appropriate choice of streamline diffusion parameters. The results ofnumerical experiments confirm our theoretical results. Moreover, they showthat despite its almost uniform convergence, a DL mesh can be fairly goodalternative to the widely used Shishkin mesh.

22

Mx = 2My ‖u− uh‖SD rate ‖uI − uh‖SD rate ‖u− uh‖∞ rate16 4.252e-01 1.31 2.371e-01 2.58 3.779e-01 2.8132 1.715e-01 1.17 3.957e-02 2.25 5.379e-02 1.6564 7.629e-02 1.09 8.334e-03 2.16 1.711e-02 1.77128 3.595e-02 1.04 1.858e-03 2.03 5.012e-03 1.87256 1.753e-02 1.02 4.539e-04 2.03 1.367e-03 1.93512 8.619e-03 1.01 1.115e-04 2.05 3.586e-04 1.971024 4.276e-03 1.01 2.700e-05 2.01 9.181e-05 1.982048 2.131e-03 - 6.691e-06 - 2.323e-05 -

Table 3: Errors on DL mesh (12)-(13) with ε = 10−8.

ε ‖u− uh‖SD ‖uI − uh‖SD ‖u− uh‖∞10−2 5.456e-04 5.495e-06 4.339e-0610−3 8.370e-04 5.172e-06 1.351e-0510−4 1.107e-03 2.071e-06 4.973e-0610−5 1.367e-03 2.785e-06 4.608e-0610−6 1.623e-03 3.893e-06 7.875e-0610−7 1.877e-03 5.197e-06 1.365e-0510−8 2.131e-03 6.691e-06 2.323e-0510−9 2.385e-03 8.374e-06 4.012e-0510−10 2.639e-03 1.025e-05 6.999e-05

Table 4: Errors on DL mesh (12)-(13) with Mx = 2My = 2048.

Acknowledgement. This paper has been supported by the Ministry ofEducation, Science and Technological Development of the Republic of Ser-bia, project no. 451-03-68/2020-14/200134 and project no. 451-03-68/2020-14/200156: ”Innovative scientific and artistic research from the FTS activitydomain”. The authors are grateful to Professor S. Franz (TU Dresden, Ger-many) for his MATLAB codes.

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DL mesh Shishkin meshMx = 2My ‖uI − uh‖SD rate ‖uI − uh‖SD rate

16 2.371e-01 2.34 4.857e-02 1.3132 2.488e-02 2.25 1.962e-02 1.4564 5.243e-03 2.20 7.170e-03 1.55128 1.137e-03 2.13 2.453e-03 1.61256 2.600e-04 2.04 8.027e-04 1.66512 6.332e-05 2.01 2.542e-04 1.701024 1.568e-05 2.01 7.846e-05 1.722048 3.893e-06 - 2.374e-05 -

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