Uniform Approximation of Singularly Perturbed Reaction ...xenophon/pubs/cx_sf.pdf · Uniform Approximation of Singularly Perturbed Reaction-Diffusion Problems by the Finite Element

Uniform Approximation of Singularly PerturbedReaction-Diffusion Problems by the FiniteElement Method on a Shishkin MeshChristos Xenophontos,1 Scott R. Fulton2

1Department of Mathematical Sciences, Loyola College, 4501 N. Charles Street,Baltimore, Maryland 21210-2699

2Department of Mathematics and Computer Science, Clarkson University,Potsdam, New York 13699-5815

Received 23 December 2001; accepted 17 June 2002

DOI 10.1002/num.10034

We consider the numerical approximation of singularly perturbed reaction-diffusion problems over two-dimensional domains with smooth boundary. Using the h version of the finite element method over appropri-ately designed piecewise uniform (Shishkin) meshes, we are able to uniformly approximate the solution at aquasi-optimal rate. The results of numerical computations showing agreement with the analysis are alsopresented. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 89–111, 2003

Keywords: singularly perturbed problem; finite element method; Shishkin mesh

I. INTRODUCTION

The numerical solution of singularly perturbed boundary value problems has received muchattention recently, and there are numerous articles and even books written on this subject (see[1–4] and the references therein). Problems of this type arise in many areas, such as fluidmechanics and heat transfer as well as problems in structural mechanics posed over thindomains. The solution of singularly perturbed elliptic problems will, in general, containboundary layers along the boundary of the domain. These layers complicate the numericalapproximation, and the method must be carefully tailored to account for their presence. Inparticular, the method must be robust in the sense that the error in the approximation should notdeteriorate as the singular perturbation parameter tends to zero.

Currently, there are two known ways to alleviate this problem in the context of the FiniteElement Method (FEM). The first is through the use of the (traditional) h version of the FEMon “specially designed” meshes, which take into account the nature of the boundary layer. One

Correspondence to: Christos Xenophontos, Department of Mathematical Sciences, Loyola College, 4501 N. CharlesStreet, Baltimore, MD 21210-2699 (e-mail: [email protected])

© 2002 Wiley Periodicals, Inc.

such example is the Bakhvalov (exponentially graded) mesh used by Blatov [5, 6], to approx-imate the solution of reaction-diffusion problems with variable coefficients by linear finiteelements. (See also [7] for a similar exponentially graded mesh strategy.) Another example isthe so-called Shishkin mesh, which is piecewise uniform [8]. It is well known that robustquasi-optimal (algebraic) error estimates can be obtained using the Shishkin mesh (see [8–11]).The second way one can effectively approximate boundary layers is through the use of the hpversion of the FEM with appropriately designed meshes (see [12–17]). The advantage of thissecond approach is the exponential rates of convergence that can be established when thedomain is smooth [12].

Our goal in this article is to extend the results of [9] for the case of general domains withsmooth boundary. In [9], a typical singularly perturbed problem was considered on the unitsquare, under certain assumptions that eliminated the presence of corner singularities. In somesense, their analysis “simulates” the case of a smooth domain, while taking advantage of therectangular nature of the problem and hence the design of the mesh. We wish to show that therobustness and uniformity of the method is preserved, even for general domains with smoothboundary.

Throughout the article, Hk(�) will denote the Sobolev space of order k on a domain � � R2,with H0(�) � L2(�), and � � �k,�, � � �k,�, denoting the norm and seminorm as usual. Also, wewill make use of H0

1(�) � {u � H1(�) : u � 0 on ��}. For I � [a, b] � R, we will write Hk(I)and H0

1(I) � {u � H1(I) : u(a) � u(b) � 0}. The letters c and C will be used to denote genericpositive constants, possibly not the same in each occurrence.

II. THE MODEL PROBLEM AND ITS REGULARITY

We consider the two-dimensional singularly perturbed problem

��2�u � u � f in � � R2

u � 0 on ��, (2.1)

where � is a small parameter that can approach zero and �� is smooth. Its variationalformulation reads: find u� � H0

1(�) such that

B�u�, v� � F�v� � v � H01�� (2.2)

where

B�u, v� � ��

��2�u � �v � uv dxdy, F�v� � ��

fv dxdy. (2.3)

For any f � L2(�), (2.2) admits a unique solution which will contain, in general, boundary layersalong the boundary of the domain. In particular, u� can be decomposed into a smooth part, aboundary layer part, and a smooth remainder, as given in Theorem 2.1 below (see also [18] forsimilar decompositions). For this decomposition, we will define boundary fitted coordinates ina region near the boundary. With �0 a positive constant less than the minimum radius of

90 XENOPHONTOS AND FULTON

curvature of �� and n�z the outward unit normal at a point z � z(� ) � (X(� ), Y(�)) � ��, wedefine

�0 � �z � �n� z : z � ��, 0 � �0. (2.4)

Then, the boundary fitted coordinates are defined by the correspondence

��, �� 3 z � �n� z � �X�� Y��, Y�� X��.

See also Fig. 1.

Theorem 2.1. [16] Suppose f � H4M�2(�) for some M � {0, 1, 2, . . .}. Then

u� � w�M � uBL

M � r�M, (2.5)

where � C�([0, �)) is a cutoff function satisfying

�� 1 for 0 � � � �0/30 for � � 2�0/3

(2.6)

with �(s)(�)� � C(�0, s), s � 0, 1, . . . ,

w�M�x, y� � �

i�0

M

�2i��i�f�x, y�, (2.7)

and in �0

uBLM � �

i�0

M

�i��ie��/� (2.8)

with �i(�) smooth and independent of �. The remainder in (2.5) satisfies

FIG. 1. Definition of �0 and boundary fitted coordinates.

FEM ON A SHISHKIN MESH 91

�r �M�k,� � C�M�3/2�k, 0 � k � M � 3/2,

where C � R is independent of �.The key observation from the above theorem is that the boundary layer effect is essentially

one-dimensional, namely in the direction normal to the boundary. Hence, if f is smooth, then thedifficulty in approximating u� lies entirely within the boundary layer term uBL

M , and in particularthe one-dimensional function

uBL�� :� �ie��/�. (2.9)

III. THE FINITE ELEMENT METHOD

With SN � H01(�) a finite dimensional subspace of dimension N, we seek u�

N � SN such that

B�u�N, v� � F�v� � v � SN. (3.1)

We have

�u� � u�N��,� � inf

v�SN

�u� � v��,�, (3.2)

where the energy norm � � ��,� is defined by

�u��,�2 � �2�u�1,�

2 � �u�0,�2 . (3.3)

The space SN will be defined as follows. Let K � (�1, 1)2 denote the reference square andlet p(K) denote the set of polynomials on K of degree � p in each variable. For a generalquadrilateral K � �(K), with � is a smooth invertible mapping, we have � p(K), if and onlyif � � ( � ��1) � p(K). If T is a triangle, p(T) denotes the set of polynomials of total degree� p over T. Let p� � (p1, . . . , pm) denote the polynomial degree vector and � � {Ki}i�1

m be somesubdivision of the domain �, which is regular in the sense of [19]. We then define

S��, �, p� � � �u � C0�� : u�Ki� pi

�Ki�, Ki � �, i � 1, . . . , m

and set SN :� SN(�, �, p�) � S(�, �, p�) � H01(�).

Let u�N � u1 � u2 � u3 � SN, for some ui � SN, i � 1, 2, 3. Then, combining (2.5) and (3.2)

we have

�u� � u�N��,� � �w�

M � u1��,� � �uBLM � u2��,� � �r �

M � u3��,�. (3.4)

Since by Theorem 2.1, w�M and r �

M are smooth, the difficulty in obtaining a uniform estimate forthe error �u� � u�

N��,� depends on the ability to bound �uBLM � u2��,�. Moreover, since uBL

M

vanishes outside of �0 we further see that

�uBLM � u2��,� � �uBL

M � u2��,�0.


In the next section we will construct the finite element space for the approximation of uBLM taking

into account the above observations and the explicit representation (2.8).

IV. THE APPROXIMATION OF THE BOUNDARY LAYERA. One Dimensional Analysis

In view of Theorem 2.1 and (2.9), we will first focus our attention to the approximation of

uBL�x� � xqe�x/�, q � 0, 1, . . . , M (4.1)

over I � (0, 1), using spaces of piecewise polynomials of degree p, defined on the so-calledShishkin mesh. The resulting mesh-degree combination will be denoted by � � (�I, p�). TheShishkin mesh is defined as follows. With p(I) the set of polynomials on I of degree less thanor equal to p, we set

� � min�12

, ��p � 1�ln�1 � n� (4.2)

for some integer n � 1. Let I � I (1) � I (2), where I (1) � (0, �) and I (2) � (�, 1), and divide I (1)

into n equally spaced subintervals of length � /n and I (2) into 2n equally spaced subintervals oflength (1 � �)/(2n) (Fig. 2). Denote the resulting (Shishkin) mesh by �I � {Ij}j�1

3n and let

S�� S��I, p� � � �u � C0�I� : u�Ij� p�Ij�, Ij � �I (4.3)

be the corresponding space of piecewise polynomials with p� � ( p, . . . , p) andN :� dim(S(�)) � O(n). The following Lemma gives the main tool for the approximationof uBL

M .

Lemma 4.1. Consider uBL(x) � xqe�x/� on I � (0, 1) and let S(�) be the space defined by (4.3)over the Shishkin mesh, with dim(S) � O(n). Then, there exists up � S(�) such that

�uBL � up��,I � C�1/2�n�1ln�n � 1��p, (4.4)

where C � R is independent of � and n, but depends on p and q.Proof. See proof of Theorem 2.2 in [11] for the case q � 0. The extension to q � 0 is

straight forward. y

B. Two-dimensional Analysis

We now return to the approximation of uBLM , given by (2.8). We will define the space SN

BL for theapproximation of the boundary layer, using the Shishkin mesh defined above. To this end, let�0 be given by (2.4) and set �1 � ��0. In practice, �0 could be selected by dividing �� into

FIG. 2. Example of one-dimensional Shishkin mesh with n � 3.


m fixed subintervals (�j, �j�1), j � 1, . . . , m, �m�1 � �1 � L, where L is the length of ��, anddrawing the inward normal at �j, j � 1, . . . , m of length �0. Then connect each point(�j, �j) � (�0, �j) using the curve � � �0 (with �0 constant) and define

�0j � ��, �� : 0 � � � �0, �j � � � �j�1, j � 1, . . . , m. (4.5)

Note that by construction �0 � �j�1m �0

j . Divide each �0j into

�0,1j � ��, �� : 0 � � � �0�, �j � � � �j�1,

�0,2j � �0

j ��0,1j

where � � (0, 1) is given by (4.2). Further divide �0,1j using n subdivisions and �0,2

j using 2nsubdivisions, each being equally spaced in the � direction. This will produce a Shishkin mesh inthe � direction over each �0

j with 3n rectangular elements. To obtain a fully two-dimensionalmesh over each �0

j , we take a “tensor-product” of the Shishkin mesh in the � direction, with auniform mesh of 2n elements in the � direction. The resulting mesh over �0

j , denoted by �0j , will

include 6n2 elements (Fig. 3). We will then use �0 to denote the mesh over �0 constructed withthis technique, which will include 6mn2 � O(n2) elements.

The space for the finite element approximation of the boundary layer will be defined “locally”in the (�, �) variables, as follows. Let

p��0� � � ��, �� : ��, �� 1��2��with �1, �2 polynomials of degree � p over �0

�and define

SNBL � �v � C0�� : v��1 � 0, v�� 0,i

j � p��0,ij �,

�0,ij � �0

j , i � 1, 2, j � 1, . . . , m � . (4.6)

We have the following theorem.

FIG. 3. Example of the Shishkin mesh over �0j with (a) n � 1, (b) n � 2, (c) n � 4.


Theorem 4.2. Let uBLM be given by (2.8), let SN

BL be the space defined by (4.6) with N :� dim(SNBL)

and let n be the number of subdivisions within �0 in the direction normal to the boundary. Thenthere exists u2 � SN

BL such that

�uBLM � u2��,� � C�1/2N�p/2�ln�n � 1��p, (4.7)

with C � R independent of � and N.Proof. First note that outside of �0, uBL

M vanishes and u2 � SNBL can be chosen to be zero

there. Thus,

�uBLM � u2��,� � �uBL

M � u2��,�0 � �uBLM � u2��,�0, � �uBL

M � u2��,�0��0, (4.8)

where �0, � { z � �n�z : z � ��, 0 � � � �0/3}. Let us first consider the approximation over�0,

j � �0,, j � 1, . . . , m, with

�0,j � ��, �� : 0 � � � �0/3, �j � � � �j�1. (4.9)

Using the change of variables

� � 3�/�0, � � �� j�/��j�1 � �j�,

we map �0,j from the �-� plane to � � (0, 1)2 in the �-� plane. Note that under this

mapping,

uBLM � �

i�0

M

�i��ie��/�

becomes

uBLM � �

i�0

M

�i��0��ie��0�/� (4.10)

with �i(�) smooth and satisfying ��i�k,� � C(k) � R. We have

�uBLM � u2��,�0,

j � C�uBLM � u2�� ,� � C� �

i�0

M

�i��0��ie��0�/� � u2�� ,�

.

Writing u2(�, �) � ¥i�0M vi(�)wi(�) � SN

BL with vi and wi polynomials of degree less than or equalto p, we further have


�uBLM � u2��,�0,

j � C �i�0

M

��i��0��ie��0�/� � vi��wi��,�

� C �i�0

M

��i��0��ie��0�/� � �i��vi�� i��vi�� vi��wi��,�

� C �i�0

M

��i��,I��0��ie��0�/� � vi��,I � �vi��,I��i�� wi��,I.

By Lemma 4.1,

��0��ie��0�/� � vi��,I � C�1/2�n�1ln�n � 1��p,

where 3n is the number of elements used in the � direction. Hence (using the regularity of �i),

�uBLM � u2��,�0

j � C��1/2�n�1ln�n � 1��p � �i�0

M

�vi��,I��i�� wi��,I�. (4.11)

We proceed by noticing that

�vi��,I � �vi�� e��0�/� � e��0�/��,I � �vi�� e��0�/��,I � �e��0�/��,I

� C�vi�� e��0�/��,I � C�1/2,

so using Lemma 4.1 we get

�vi��,I � C�1/2�n�1ln�n � 1��p � C�1/2. (4.12)

We next use standard finite element estimates (see [19]), to get

��i�� wi��,I � Cn�p. (4.13)

Combining (4.11)–(4.13), we get

�uBLM � u2��,�0

j � C�1/2�n�1ln�n � 1��p � C��1/2�n�1ln�n � 1��p � �1/2�n�p

� C�1/2�n�1ln�n � 1��p.

Since j � 1, . . . , m with m fixed, we have

�uBLM � u2��,�0, � C�1/2�n�1ln�n � 1��p

so that (4.8) becomes

�uBLM � u2��,� � C�1/2�n�1ln�n � 1��p � �uBL

M � u2��,�0��0,. (4.14)


We finally consider the approximation over �0��0, and bound the last term in (4.14). Usingstandard finite element estimates (see [19]) we get

�uBLM � u2��,�0��0, � CN�p/2�uBL

M �p�1,�0��0,,

where N � dim(SNBL). By assumption �(s)(�)� � C(�0, s), hence

�uBLM �p�1,�0��0, � C��0, p��uBL

M �p�1,�0��0, � C��0, p��p�1exp��0/��.

This gives

�uBLM � u2��,�0��0, � C��0, p��p�1N�p/2exp��0/�� C��0, p�N�p/2�1/2

exp��0/��

�p�3/2 ,

hence

�uBLM � u2��,�0��0, � C�1/2N�p/2, (4.15)

with C independent of N and �, but depending on p and �0. Combining (4.14) and (4.15) we get

�uBLM � u2��,� � C�1/2�n�1ln�n � 1��p � C��0, p��1/2N�p/2.

Since N � O(n2), we have n�p/2 � CN�p/2 and

�uBLM � u2��,� � C��0, p��1/2�N�p/2�ln�n � 1��p � N�p/2

from which the result follows. y

V. THE APPROXIMATION OF THE SMOOTH PART AND THE REMAINDER

In this section we will discuss the approximation of the smooth part w�M, and the remainder r�

M.Recall that

w�M�x, y� � �

i�0

M

�2i��i�f�x, y� (5.1)

and

�r �M�k,� � C�M�3/2�k, 0 � k � M � 3/2, (5.2)

where C � R is independent of �. Since by assumption f is smooth, w�M is also smooth. By

defining the space

SNw,r � �v � C0�� : v��j

� p��j�, �j � �, (5.3)


with � � {�j} a quasi-uniform mesh, we have by the standard finite element theory (see e.g.,[19]) that there exist u1, u3 � SN

r,w such that

�u1 � w�M��,� � CN�p/2�w�

M�k,�, k � p � 1

�u3 � r �M��,� � CN�p/2�r �

M�k,�, p � 1 � k � M � 3/2 (5.4)

where N � dim(SNr,w) and C � R is independent of � and N. Therefore,

�u1 � w�M��,� � CN�p/2� �

i�0

M

�2i��i�f�x, y��k,�

� CN�p/2 �i�0

M

��2i��i�f�x, y��k,�.

Since f is smooth,

�u1 � w�M��,� � C�M�N�p/2. (5.5)

Similarly, combining (5.2), (5.4) we see that

�u3 � r �M��,� � C�1/2N�p/2. (5.6)

VI. THE APPROXIMATION OF U�

We now combine the results obtained thus far, to approximate the solution u� � w�M � uBL

M �r�

M to (2.2). Define the space SN � H01(�) as1

SN � �u � C0�� : u��j� p��j�, �j � �, (6.1)

where � is chosen as follows. In �0 we will use the mesh �0 from Section IV.B., whereas in�1 � ��0 we will use a quasi-uniform mesh that is compatible with �0. An example of thecombination of the two meshes over a circle is shown in Figure 4.

Note that the space SN is the usual conforming finite element space of piecewise polynomialson a piecewise quasi-uniform mesh and SN

BL � SN. Hence u � (u1 � u2 � u3) � SN satisfy (4.7),(5.5), and (5.6). We are now in a position to present our main result.

Theorem 6.1. Let u� be the solution to (2.2) as given by (2.5) and let vN � SN be its finiteelement approximation, where the space SN is defined by (6.1) with N :� dim(SN) and n as inTheorem 4.2. Then,

�u� � vN��,� � CN�p/2�ln�n � 1��p, (6.2)

with C � R independent of � and N.Proof. We have, using (3.4), (4.7), (5.5), and (5.6),

1 We could (equivalently) define SN as the union of SNBL and SN

r,w given by (4.6) and (5.3), respectively.


�u� � vN��,� � ��w�M � uBL

M � r �M� � �v1 � v2 � v3��,�

� �w�M � v1��,� � �uBL

M � v2��,� � �r �M � v3��,�

� CN�p/2 � C�1/2N�p/2�ln�n � 1��p � C�1/2N�p/2

from which the result follows. y

Corollary. Let the assumptions of Theorem 6.1 hold. If the right hand side of (2.1)f( x, y) � SN, then

�u� � vN��,� � C�1/2N�p/2�ln�n � 1��p, (6.3)

with C � R independent of �.

FIG. 4. Example of the mesh over a circle, with p � 1, � � 0.05, and n being the number of subdivisionsin the boundary layer region.

FIG. 5. Convergence for the first example ( f � 1) with p � 1.


Proof. If f � SN, then w�M � SN as well; hence, �w�

M � v1��,� � 0. The result follows asin the proof of Theorem 6.1 above. y




VII. NUMERICAL RESULTS

In this section we present the results of numerical computations for the problem (2.1), in the casewhen � is the unit disk. We choose this domain for our computations in order to be able to easily

FIG. 8. FE solution (top) and error (bottom) for the first example ( f � 1) with p � 1.


construct exact solutions for comparison purposes. We will be plotting the percentage relativeerror in the energy norm

Error � 100 ��uEX � uFEM��

�uEX��

FIG. 9. FE solution (top) and error (bottom) for the first example ( f � 1) with p � 2.


versus N/[ln(n � 1)]2 in a log-log scale.2 Here N is the number of degrees of freedom, n is thenumber of subdivisions within the boundary layer region in the direction normal to the boundary(cf. Theorem 4.2), uEX denotes the exact solution, and uFEM denotes the finite element solution.The goal is to verify the robustness and convergence rate of the method, as predicted byTheorem 6.1 and (6.3). The computations were performed using the same mesh as in Fig. 4, forseveral values of � and with piecewise polynomials of degree p � 1, 2, and 3.

First, we choose the right hand side in (2.1) to be f � 1 (i.e., f � SN). Figures 5–7 show thatthe method exhibits the expected robustness and converges at the quasi-optimal rate predictedby (6.3). Note that the presence of the term �1/2 in the error bound (6.3) is visible in practice andthe method not only does not deteriorate as � 3 0 but actually performs better. In Figs. 8 and9 we plot the finite element solution as well as the pointwise error, for p � 1 and p � 2,respectively; the remaining parameters were chosen as � � 0.01, n � 5. (Other choices of � andn produce similar results.) The plots show that the error is higher in the boundary layer region,an observation made in [9] as well.

For our second example, we choose f to be such that the exact solution is given, in polarcoordinates, by

u�r, �� u�r� � cos��r2� �I0�0, r/��

I0�0, 1/��, (7.1)

where I0(r) is the modified Bessel function of order 0. Even though I0(r) is not the typicalboundary layer function (2.9), its behavior is very similar (see e.g., [20]). Figures 10–12 show

2 The error curves will (eventually) be straight lines with slope �p/2.

FIG. 10. Convergence for the second example (7.1) with p � 1.


once more the robustness of the method (as predicted by Theorem 6.1) and in addition, eventhough f � SN, the method performs better as �3 0, which suggests that perhaps the estimate(6.3) is valid in a more general setting. In fact, it is not difficult to see from (5.1) that if the




lowest order term in the smooth part of the asymptotic expansion for u includes a positive powerof �, then an error estimate similar to (6.3) will hold in this case, even if f � SN. (The figuressuggest that the error bound will have a factor of ��, for some � � 0.) In Figs. 13 and 14 we

FIG. 13. FE solution (top) and error (bottom) for the second example (7.1) with p � 1.


plot the finite element solution as well as the error, corresponding to p � 1, 2, for the case� � 0.01, n � 5.

We also note from the examples above that when � � 0.1 and p � 3, the rate of convergenceappears to be better than the one predicted. This is due to the fact that for large � the solution

FIG. 14. FE solution (top) and error (bottom) for the second example (7.1) with p � 2.


is “smooth-enough” so that using “high” p results in an initially faster rate of convergence; thepredicted rate will be visible as N 3 �.

As a final numerical example we consider the following generalization of (2.1):

FIG. 15. Convergence for the third example (7.3) with p � 1.



��2�u � au � f in � � R2

u � 0 on �� (7.2)

with � the unit disk as before and a � a(r, � ) � r2cos(2�) (in polar coordinates). The right handside f in (7.2) was chosen so that the exact solution is given, in polar coordinates, by

u�r, �� r cos��1 �I0��r/��

I0��1/��. (7.3)

We repeat the same computations and present the results in Figs. 15–19. Once again the methodconverges uniformly (in �) at a quasi-optimal rate similar to (6.3). As in the previous twoexamples, we see that the rate is a little higher for � � 0.1 when p � 3 and in addition, forp � 2 the errors increase for small N (since the subspaces are not strictly nested) but eventuallyapproach the conjectured asymptotic rate. These numerical results suggest that the method canpossibly be extended to the variable coefficient case even though at present our analysis does notapply. In fact, the only ingredient needed to extend our results to this case would be an analogof Theorem 2.1, since the approximation theory is easily extendable, and these numericalexperiments suggest that this is possible. This will be the focus of a future communication.

VIII. CONCLUSIONS

The uniform approximation of singularly perturbed problems is a challenging task. The use ofspecially designed meshes is necessary for the uniform approximation of such problems by the



finite element method. One example is the Shishkin mesh, which is piecewise uniform. Usingthe Shishkin mesh in the direction normal to the boundary of the domain, we establishedquasi-optimal convergence rates that are uniform with respect to the perturbation parameter, in

FIG. 18. FE solution (top) and error (bottom) for the third example (7.3) with p � 1.


the case of two-dimensional domains with smooth boundary. Referring to the work of Kelloggand Stynes [21], the rates obtained here are optimal up to the logarithmic factor [ln(n � 1)]p.The results of our numerical computations not only agreed with the analysis, but suggested that

FIG. 19. FE solution (top) and error (bottom) for the third example (7.3) with p � 2.


the method can be extended to variable coefficient singularly perturbed elliptic boundary valueproblems.

References

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