Supported by Beijing Natural Science Foundation (No. 1122014).
The uniform convergence of upwind schemes on
layer-adapted meshes for a singularly perturbed Robin BVP
Quan Zheng, Fengxi Huang, Xiaoli Feng, Mengbin Han
College of Sciences, North China University of Technology, Beijing 100144, China
Abstract—In this paper, we discuss the uniform convergence of the simple upwind scheme on the Shishkin mesh and the
Bakhvalov-Shishkin mesh for solving a singularly perturbed Robin boundary value problem, and investigate the midpoint upwind
scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh to achieve better uniform convergence. The elaborate ε
-uniform pointwise estimates are proved by using the comparison principle and barrier functions. The numerical experiments
support the theoretical results for the schemes on the meshes.
Keywords—Singularly perturbed Robin BVP; simple upwind scheme; midpoint upwind scheme; layer-adapted mesh; uniform
convergence
1. Introduction
Let us consider a singularly perturbed convection-diffusion
Robin boundary value problem:
01
()()(), (0,1)
(0) ,(1) (1) ,
Lu ubxucxufxx
Buu ABuuuB

   


 

(1)
where 01

 is a small perturbation parameter, A and B are
given constants, and function)(),( xcxband )(xf are
sufficiently smooth with 0)(0


xb and 0)( xc .
Under these conditions, the singularly perturbed problem (1)
has a unique solution with a boundary layer at 0x

.
Singularly perturbed problems arise in many branches of
science and engineering such as modeling fluid flows and
simulating semiconductor devices (see [1-4]). A wide variety
of numerical methods, including the simple upwind scheme
and the midpoint upwind scheme on layer-adapted meshes,
were constructed to solve the problems in the past few decades
(see [5-9]).
In this paper, the properties of the exact solution and the
Shishkin mesh are introduced in section 2. In section 3, we
discuss the simple upwind scheme on the Shishkin mesh for
solving the singularly perturbed Robin BVP (1) and prove its

-uniform pointwise convergence of order )( 1 NO on the
nodes in coarse part and)ln( 1NNO on the nodes in fine part.
In section 4, the simple upwind scheme on the
Bakhvalov-Shishkin mesh, and the midpoint upwind scheme
on the Shishkin mesh and the Bakhvalov-Shishkin mesh are
studied to reach higher orders of uniform convergence. In
section 5, several numerical examples support the elaborate
error estimates.
2. The Solution and the Mesh
Lemma 1 (see [5]) For any positive integer0q, if )(xu
is the solution of problem (1) with sufficiently smooth data
then ()ux can be decomposed as uSE , where the
smooth part S satisfies
() ()
L
Sxf x

and ()
() ,
i
Sx C 0,iq
while the part E satisfies 0)(xLE ,
and )exp()( 1)(



x
CxEi, 0iq .
Let }
ln2
,
2
1
min{



N
, N be an even positive number ,
and

be the transition point, where 1 N 

 as generally in
practice. We have the Shishkin mesh:
2,0 ,
2
() 2(1)()
2,,
2
N
ii
N
xi N
iNiN
N










(2)
which is simply piecewise equidistant. Denoting 1iii
hxx


,
we have
Lemma 2. N
N
hi


ln4
, 11
/2 2 Ni
Nh N


 ,
1, 2,...,/2.iN

Throughout the paper, C is a generic positive constant that is
independent of

and i h , and note that Ccan take
different values at each occurrence, even in the same
argument.
Open Journal of Applied Sciences
Supplement：2012 world Congress on Engineering and Technology
66

3. The Scheme and Its Estimate
For the simple upwind scheme:
00 01
1, 2,...,1,
,,
NNNN N
iiiiiii
NN NNNNN
NN N
L
uDDubDucuf
iN
Bu uABu uDuB

 

 


 

(3)
where
11
1
,,
NN NN
NN ii ii
ii
ii
uu uu
Du Du
hh





1
2() ,
NN
Nii
i
ii
Du Du
DDuhh





we have
1 1
111111
11
22 2
()( )
() ()
,
N
NN NN ii
ii iii
iiiiiiiii i
NcN N
iiii ii
bb
L
uu cuu
hhhhhhhhh h
rururu


 



 

 
and ,, 0,
c
iii
rrr

 0,
c
iii i
rrrc

  1,2,...,1.iN
Lemma 3. If grid function ()
i
vx and ()
i
wx satisfy
000 0
BB
NN
vw, 11 BB
NN
N
N
vw and NN
ii
Lv Lw, 1, 2,...,1iN
,
then ()()
ii
vx wx, 0,1, 2,...,iN, and the equation (3) has a
unique solution.
Proof. It is proved by that the coefficient matrix associated
with N
L is an M-matrix.
By direct computation and Taylor formulas as usual, we
have the following two lemmas.
Lemma 4. If 


i
j
j
i
h
ZZ
1
0)
2
1(,1


,
then },max{ 1

i
i
i
N
h
CZ
ZL

, 1, 2,...,1iN .
Lemma 5. 1
11
()[ ()()]
ii
ii
xx
NN
ii xx
L
uu Cutdtutdt 


   

.
As in the continuous case, decompose the numerical
solution into the smooth part and the layer part by
N
NN
iii
uSE , we have NN
ii
L
Sf

, 0,1, 2,...,1iN ,
00 (0)
NN
BS S,
1
1(1) (1)
NN
NNNNN
NN
N
SS
BS SSS
h


 ,
and 0
N
i
NEL , 1,...,1,0  Ni, )0(
00EEB NN,
1
1(1) (1)
NN
NNN NN
NN
N
EE
BE EEE
h


 .
Therefore, the error can be estimated by
.
NN N
iiiiii
uu SSEE
For the smooth part, we have 00 0
()0,
NN
BS S
1
111
()() ,
NNN
NNN N
BS SBSBSCN


1
111
()(),
NNN
iii i
L
SSLS LSCN

 
for ,1,...,2,1


Ni by Lemma 1 and Lemma 5. Setting
)3(
1
ii xCN  
 for all i, we have 1
CNL i
N

)( N
ii
NSSL . By the discrete comparison principle, we get
1N
iii
SS CN 

 , 0,1, 2,...,.iN (4)
For the layer part, we have
Lemma 6. There exists a constant C such that
1N
ii
EE CN

 , /2,...,.iNN
Proof. By Lemma 1, we have CEEBNN  |)0(||| 00 and
12
1(1)(1) (1)
NN
N
BE EECe Ce







  
1
2
11
(1) .
2
j
h
NN
j
jj
h
Ce C



 



Let NiiZCZCY


2
00  and 0
C to be sufficiently large,
then 0000 0
N
NN
BY CBE , 101
N
NN
N
NN
BY CZBE and
0
NNN
ii
LY LE , 1, 2,...,1iN

. So, i
Y is a discrete barrier
function for N
i
E, and noting
/2 /2
2
1 1
1
ln(1)(())
2222
NN
jjj
j j
hhh



 
 


2
/2 /2
2ln ,
22
NN
xx NC
N




/2
1
/2
1
2
1
(1 ),
2
N
x
Nj
j
hCe CN






 

1
1
/2 1/2 1
2
(1 )(1)2,
2222
14
NN
j
jN jN
h
N








 




11
/2
1
11
(1 )2(1 ),
22 2
NN
jj
jj
hh
CN


 





we have
1
2/

 CNYYE Ni
N
i, NNi,...,2/.
From Lemma 1, we have
1
i
E
Ce CN




, / 2,...,iN N.
Thus, the proof is complete.
Lemma 7. There exists a constant C such that
1ln N
ii
E
ECNN

 , 1,...,/ 2iN.
Proof. By Lemma 5, Lemma 1, the mesh generating
function (2) and noting that
NCNNN
hiln)ln2sinh()
2
sinh( 11  


, 1,...,/ 2iN,
67

we have
1
11
()[ ()()]
ii
ii
xx
NN
ii xx
LEECEx dxEx dx 


 
 

1
1
21
exp( )exp()sinh(),
222
i
i
xii
x
x
h x
CdxC












1
11 11
11
lnexp()ln(1) .
22
ii
jj
jj
hh
CN NCN N




 




Let 1
0ln (1)
ii
CNN Z 


, from Lemma 4 and 6, we have
11
0ln( )
NNN
iiii
LCNNZLEE 


,
000
0
N
EE   , 1
/2 0/2/2
N
NNN
CN EE 


,
provided that the constant 0
C is chosen sufficiently large. So,
1ln
N
ii i
EECN N 

 by a discrete comparison principle.
Theorem 1. The simple upwind scheme on the Shishkin
mesh for the singularly perturbed Robin boundary value
problem (1) satisfies:
1
1
ln, 0,
2
,.
2
N
ii
N
CN N i
uu N
CNiN










(5)
Proof. It is proved by (4), Lemma 6 and 7.
I. FURTHER RESULTS
On the Bakhvalov-Shishkin mesh (see [8]):
(),
iii
i
xxttN 
,0,1, 2,...,iN,
where the mesh generating function is as follows:
211
ln(12(1)), 0,
2
() 2ln2ln11
2(1)(),1,
22
tt
N
xt NN
tt




 



 


(6)
the simple upwind scheme for solving the singularly perturbed
Robin BVP was proved to be uniform first-order
convergence (see [4]):
1N
ii
uu CN

 ,0iN. (7)
Further, we consider the midpoint upwind scheme for
Dirichlet BVP in [9] to be modified for the Robin BVP (1) as
follows:
1
111
222
11
222
00 01
, 1,2,...,1,
2
2
()()(),
22
,,
NN
NNNNii
iii
ii i
NNN NN NN
NNNNNN
NN NNN
NN NNNNN
NN N
uu
LuDDub DucfiN
bb
Luucuuf
hh hhh
Bu uABu uDuB


 
 


 



  




(8)
where 11
()/(2)
NNN
NNN N
D
uuu h

 . For 1, 2,...,1iN
, we have
1/ 21/ 21/ 21/ 2
1 1
111 111
11
22 2
()( )
()2 ()2
,
N
NN NN
ii ii
ii ii
iiiiiiiii i
NcNN
iiii ii
bc bc
Lu uuu
hhhhhhhhh h
rururu


 


 


 


Supposed that  chi/2 0 
, we have ,, 0,
c
iii
rrr


0
2/1

ii
c
ii crrr , 1, 2,...,1iN
.
For iN

, from (8), we have
1
22
222 2
()().
NN NN
NNNNNNN
NNN N
Luucb ufbB
hhh h




The coefficient matrix associated with this N L is also an
M-matrix and a discrete comparison principle holds. By using
barrier functions, we can obtain the same error estimate on the
Shishkin mesh for Robin BVP as that for Dirichlet BVP in the
following: 1
2
ln, 0,
2
,.
2
N
ii
N
CN N i
uu N
CNi N










(9)
Moreover, we can prove that the midpoint upwind scheme
on the Bakhvalov-Shishkin mesh has the uniform
convergence: 1
2
,0 ,
2
,.
2
N
ii
N
CN i
uu N
CNi N










(10)
4. Numerical Examples
The numerical results in tables 1 and 2 agree with the error
estimates for the simple upwind scheme on the Shishkin mesh
and the Bakhvalov-Shishkin mesh, denoted by S-S and S-BS,
and the midpoint upwind scheme on the Shishkin mesh and the
Bakhvalov-Shishkin mesh, denoted by M-S and M-BS. The
numerical convergence rates are computed by
)max/(maxlog 2
2
N
ii
N
ii uuuu  on the coarse part and the fine
part, respectively. Denoting the error estimate by
)(|| NCuu N
ii
  , the constant is computed by
max/ ( )
N
ii
uu N  .
Problem 1. 0, 01,
(0)0, (1)(1)1.
yyy x
yyy




 




The exact solution of this problem is
1212
12
() ()/[(1)(1)]
mxm xmm
yxeem eme, where
12
,(1(14))/(2)mm

  .
TABLE 1. THE ERRORS FOR PROBLEM 1 WITH 6 10   AND 75.0


68

N /2iN rate cons
t
/2iN ra
t
e cons
t
S-S:
64 0.0069 0.59 .107 0.0077 0.98 .492
128 0.0046 0.67 .122 0.0039 1.04 .496
256 0.0029 0.77 .133 0.0019 0.96 .498
512 0.0017 0.79 .140 9.746e-4 1.00 .499
1024 9.851e-4 0.82 .146 4.878e-4 1.00 .500
2048 5.565e-4 0.85 .149 2.440e-4 1.00 .500
S-BS:
64 0.0055 0.97 .353 0.0077 0.98 .492
128 0.0028 1.00 .362 0.0039 1.04 .496
256 0.0014 1.00 .366 0.0019 0.96 .498
512 7.170e-4 0.97 .367 9.746e-4 1.00 .499
1024 3.590e-4 1.00 .368 4.878e-4 1.00 .500
2048 1.796e-4 1.00 .368 2.440e-4 1.00 .500
M-S:
64 0.0103 0.71 .660 1.497e-5 2.00 .613e-1
128 0.0063 0.77 .809 3.744e-6 2.00 .613e-1
256 0.0037 0.82 .956 9.366e-7 2.00 .111e-1
512 0.0021 0.81 1.10 2.344e-7 2.00 .985e-2
1024 0.0012 0.85 1.23 5.874e-8 1.99 .889e-2
2048 6.663e-4 0.87 1.36 1.475e-8 1.99 .812e-2
M-BS:
64 0.0055 0.97 .350 1.497e-5 2.00 .613e-1
128 0.0028 1.00 .354 3.746e-6 2.00 .614e-1
256 0.0014 1.00 .356 9.373e-7 2.00 .614e-1
512 6.981e-4 1.00 .357 2.347e-7 2.00 .615e-1
1024 3.496e-4 1.00 .358 5.882e-8 1.99 .616e-1
2048 1.749e-4 1.00 .358 1.478e-8 1.99 .619e-1
Problem 2. 12,01,
(0)1, (1)(1)0.
yyxx
yyy

 
 




Its exact solution is given by
11
2
11
()[(54)(1 )4(1)]/[1(1 )](2).
x
yxeeex xx
 



 
  
TABLE 2. THE ERRORS FOR PROBLEM 2 WITH 6 10   AND 0.5


N /2iN rate cons
t
/2iN rate cons
t
S-S:
16 0.8961 0.64 5.170 .3594 0.97 5.75
32 0.5749 0.63 5.310 .1836 0.98 5.87
64 0.3706 0.70 5.700 .0928 0.99 5.94
128 0.2280 0.77 6.010 .0466 0.99 5.97
256 0.1336 0.80 6.170 .0234 1.00 5.98
512 0.0764 0.84 6.270 .0117 0.99 5.99
S-BS:
16 0.7088 0.90 11.30 .3594 0.97 5.75
32 0.3796 0.93 12.10 .1836 0.98 5.87
64 0.1986 0.97 12.70 .0928 0.99 5.94
128 0.1017 0.98 13.00 .0466 0.99 5.97
256 0.0515 0.99 13.20 .0234 1.00 5.98
512 0.0259 0.99 13.30 .0117 0.99 5.99
M-S:
16 0.6784 0.56 10.9 6.081e-8 3.37 .561e-5
32 0.4614 0.54 14.8 5.902e-9 3.92 .174e-5
64 0.3174 0.69 20.3 3.895e-10 4.13 .384e-6
128 0.1974 0.73 25.3 2.231e-11 4.18 .753e-7
256 0.1191 0.79 30.5 1.228e-12 2.25 .145e-7
512 0.0689 0.82 35.3 2 .576e-13 2.00 .108e-7
M-BS:
16 0.4455 0.84 7.13 2.070e-7 2.16 .530e-4
32 0.2488 0.91 7.96 4.646e-8 2.52 .476e-4
64 0.1320 0.95 8.45 8.078e-9 2.74 .331e-4
128 0.0681 0.98 8.71 1.205e-9 2.87 .197e-4
256 0.0346 0.99 8.85 1.649e-10 2.92 .108e-4
512 0.0174 0.98 8.93 2.175e-11 2.98 .570e-5
Form the theoretical analysis and the numerical results, we
conclude that S-S, S-BS, M-S and M-BS are robust, efficient
and

-uniform convergent.
REFERENCES
[1] J.J.H. Miller, R.E. O'Riordan, G.I. Shishkin, Fitted Numerical
Methods for Singular Perturbation Problems. World Scientific,
Singapore, 1996.
[2] H.-G. Roos, M. Stynes, L. Tobiska, Robust Numerical Methods
for Singularly Perturbed Differential Equations,
Springer-Verlag, Berlin, Heidelberg, 2008.
[3] S. Natesan, N. Ramanujam, An asymptotic-numerical method
for singularly perturbed Robin problems-I, Appl. Math. Comput.
126 (2002) 97-107.
[4] Z.-D. Cen, H.-Y. Gao, Uniform convergence analysis of a
singularly perturbed Robin problem on a Bakhvalov-Shishkin
mesh, J. Zhejiang Univ. (Science Edition) 31 (2004) 373-375
(in Chinese).
[5] R.B. Kellogg, A. Tsan, Analysis of some difference
approximations for a singular perturbation problem without
turning points, Math. Comput. 32 (1978) 1025-1039.
[6] A.S. Bakhvalov, On the optimization of methods for solving
boundary value problems in the presence of boundary layers, Zh.
Vychisl. Mat. Mat. Fiz. 9(1969) 841-859 (in Russian).
[7] G.I. Shishkin, Grid approximation of singularly perturbed
elliptic and parabolic equations, Second doctorial thesis,
Keldysh Institute, Moscow, 1990 (in Russian).
[8] T. Linβ, An upwind difference scheme on a novel Shishkin-type
mesh for a linear convection-diffusion problem, J. Comput.
Appl. Math. 110 (1999) 93-104.
[9] M. Stynes, H.-G. Roos, The midpoint upwind scheme, Appl.
Numer. Math. 23 (1997) 361-374.
69