Download Numerical Analysis of a Strongly Coupled System of Two

Numerical Analysis of a Strongly Coupled
System of Two Convection-Diffusion
Equations with Full Layer Interaction
Hans-G. Roos and Christian Reibiger
Strong coupling of convection-diffusion equations causes interactions between
boundary layers that are not fully understood so far. Under certain assumptions,
a new approach presented explains what happens.
1 Introduction
Let us consider the following system of convection-diffusion equations:
−εu001 − b11 u01 − b12 u02 + a11 u1 + a12 u2 = f1 ,
(1.1a)
−εu002
(1.1b)
−
b21 u01
−
b22 u02
+ a21 u1 + a22 u2 = f2 ,
subject to the boundary conditions
u1 (0) = u2 (0) = u1 (1) = u2 (1) = 0.
Introducing the corresponding matrices B and A and vectors u and f the system takes the
form
Lu B −εu00 − Bu0 + Au = f.
We assume all coefficients and the right hand sides f1 , f2 to be sufficiently smooth and
consider the singularly perturbed case 0 < ε ¿ 1.
For systems with weak coupling, i.e., b12 = b21 = 0, there exist bounds for derivatives
which show the weak interaction of layers, see [2]. However, in the case of strong coupling
the two existing papers [3, 6] do not explain all possible phenomena. In a recent survey on
systems [4], the authors state: ”Strong coupling causes interactions between boundary layers
that are not fully understood at present”. The new approach presented here will fully clarify
the situation if particular assumptions on the data are satisfied.
The behaviour of the solutions of (1.1) strongly depends on the eigenvalues of the matrix
B. If the eigenvalues of B have the same sign at either x = 0 or x = 1, for instance, both
eigenvalues are positive, then u1 and u2 form overlapping layers at x = 0. For example, a
special case of this type is studied in [6]. The assumptions made in [6]
b11 > 0, b22 > 0, b12 ≤ 0, b21 ≤ 0, b11 + b12 ≥ α > 0, b21 + b22 ≥ α > 0
imply that the minimal eigenvalue of B satisfies λmin ≥ α. Consequently, overlapping layers
will form at x = 0.
The more interesting case is characterized by eigenvalues of different sign. In the paper
[3] this case is studied too but Linß only proves rough estimates for derivatives. He solely
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conjectures about the possible layer structure. Moreover, Linß pointed out that in some
cases it is not clear how the solution of the reduced problem looks like.
In the paper we shall present the layer structure and precise bounds for derivatives assuming
(V1 ) B is symmetric.
(V2 ) A + 1/2 B 0 is positive semidefinite.
(V3 ) The eigenvalues of B satisfy |λ1,2 | > α > 0 for all x.
Note that if B is not symmetric, but b12 6= 0 and b21 6= 0, one could symmetrize the problem
multiplying the second equation by b12 /b21 .
2 Asymptotic approximations and bounds for derivatives
We are particularly interested in the case that the eigenvalues of B(x) have different signs.
Therefore we assume
λ1 (x) > 0,
λ2 (x) < 0 for all x ∈ [0, 1].
(2.1)
Later we shall also comment on the case when λ1,2 > 0, which was studied in [6].
First, we prove the existence of a weak solution of problem (1.1), namely
ε(u0 , v 0 ) + (−Bu0 + Au, v) = (f, v) ∀ v ∈ (H01 (0, 1))2 .
Introduce the L2 (0, 1) norm and the H k (0, 1) seminorm (k = 1, 2, · · · ) by
kf k20 B kf1 k20 + kf2 k20
and |u|2k B |u1 |2k + |u2 |2k .
The symmetry of B allows to apply the standard technique for weak formulations of boundary
value problems. This results in
Lemma 2.1
Assume (V1 ) and (V2 ) hold. Then problem (1.1) admits a unique weak solution u in the space
(H01 (0, 1) ∩ H 2 (0, 1))2 which satisfies the a priori estimate
ε3/2 |u|2 + ε1/2 |u|1 ≤ kf k0 .
(2.2)
Next we construct asymptotic approximations for u. These approximations clarify as well
the layer structure as the definition of the reduced problem. The remainder term in the
asymptotic approximation will be estimated using (2.2).
The assumption (V3 ) guarantees the existence of B −1 . Consequently, there exists a solution
w0 of the reduced equation
−Bw0 + Aw = f.
(2.3)
Further, we denote by wh the fundamental matrix of the homogeneous reduced equation
−Bw0 + Aw = 0.
We can choose the two vectors (wh,11 , wh,12 )T and (wh,21 , wh,22 )T forming wh in such a way
that wh (0) = E (here E denotes the identity matrix).
Next we ask for the existence of layers at, for instance, x = 0 and introduce the local
variable ξ := x/ε. With respect to ξ the layer function should satisfy the constant coefficient
problem
d2 v
dv
+ B(0)
= 0.
2
dξ
dξ
2
Our assumptions yield the existence of a positive eigenvalue λ̃1 of B(0) and thus the existence
of an exponentially decreasing layer function. Denote the related eigenvector of B(0) by
[α1 , β1 ]T . Analogously, the layer at x = 1 is characterized by the negative eigenvalue λ̃2 of
B(1) and the corresponding eigenvector [α2 , β2 ]T . We set
· ¸
· ¸
α
α
u0as B w0 + wh c̃0 + d1 1 exp(−λ̃1 x/ε) + d2 2 exp(λ̃2 (1 − x)/ε).
(2.4)
β1
β2
Here c̃0 is an unknown vector and d1 , d2 are unknown constants. They are determined by
the boundary conditions:
· ¸
· ¸
α
α1
+ d2 2 exp(λ̃2 /ε) = 0,
w0 (0) + c̃0 + d1
β2
β1
· ¸
· ¸
α
α1
exp(−λ̃1 /ε) + d2 2 = 0.
w0 (1) + wh (1)c̃0 + d1
β2
β1
Because the exponential terms are exponentially small, elimination of c̃0 shows: the system
has an unique solution if
(V4 )
α2 (α1 wh,21 (1) + β1 wh,22 (1)) − β2 (α1 wh,11 (1) + β1 wh,12 (1)) 6= 0.
If the matrix B is a constant matrix and A = 0, condition (V4 ) follows automatically from
our assumptions.
Introducing c0 = limε→0 c̃0 , we observe that the reduced solution w0 + wh c0 does not
satisfy any boundary conditions, in general. In the generic case, both solution components,
u1 and u2 , exhibit boundary layers at x = 0 and x = 1. The exceptional case is characterized
by the occurrence of a zero in either component of the eigenvectors above.
Example 2.2
Let us study the system
·
¸
· ¸
−1 1 0
1
−εu −
u =
0 3
2
00
(2.5)
with homogeneous boundary conditions. The same system with two small parameters was
studied in [3]. Our asymptotic approximation is
·
¸
· ¸
· ¸
x/3
1
1
0
uas B
+ c̃0 + d1
exp(−3x/ε) + d2
exp(−(1 − x)/ε).
−2x/3
4
0
The unknowns c0 , d1 , d2 solve the system
· ¸
·
¸
· ¸
1
1/3
1
c0 + d1
= 0, c0 +
+ d2
= 0.
4
−2/3
0
Thus, for the reduced solution we obtain
u1,red = x/3 + 1/6,
u2,red = −2x/3 + 2/3.
Its first component does not satisfy the boundary conditions. Therefore, u1 has layers at
x = 0 and x = 1. The second component, however, has to satisfy the boundary condition at
x = 1 because β2 = 0 (exceptional case). u2 has only a strong layer at x = 0.
Remark 2.3
In the classical paper [1] the authors consider a symmetric matrix B with a block structure.
For our system of two equations this corresponds to the case
·
¸
b11 0
B=
0 b22
3
with negative b11 and positive b22 . Now the eigenvectors are (1, 0)T and (0, 1)T . Therefore,
u1 has only a strong layer at x = 1 and u2 at x = 0. The possible coupling via the matrix
A can, additionally, lead to weak layers of u1 at x = 0 and of u2 at x = 1.
Remark 2.4
When both eigenvalues of B are positive, our asymptotic approximation takes the form
u0as
· ¸
· ¸
α1
α
B w0 + wh c̃0 + d1
exp(−λ̃1 x/ε) + d2 2 exp(−λ̃2 x/ε).
β1
β2
Then w0 + wh c0 has to satisfy the boundary conditions at x = 1 and is, consequently, the
solution of an initial value problem for the reduced equation.
u1 and u2 do have two overlapping layers at x = 0. This fact easily explains the ”nonmonotone” layer behaviour observed without detailed explanation in [6].
Finally we bound the remainder of our asymptotic approximation. This leads to a decomposition of the solution into a ,,smooth” part and layer components. Set
u = u0as + R0 .
The remainder R0 solves a problem of the same type as u:
LR0 = g0 ,
R0 |x=0 = R0 |x=1 = 0.
Our construction of R0 yields kg0 k0 ≤ Cε1/2 . Remark, that the layer correction gives only
terms of order O(1) in g0 if measured in the maximum norm. In the L2 norm, however, we
gain ε1/2 . Then Lemma 1 gives
kR0 k∞ ≤ |R0 |1 ≤ C.
That means we proved so far: u allows a decomposition
u = S + E1 + E2
with kSk∞ ≤ |S|1 ≤ C,
where the layer terms satisfy
(k)
|E1 | ≤ Cε−k exp(−λ̃1 x/ε),
(k)
|E2 | ≤ Cε−k exp(−λ̃2 (1 − x)/ε).
For proving optimal error estimates for linear finite elements on layer adapted meshes, we
need information on |S|2 in a corresponding decomposition. Therefore, in the next step we
increase the order of our asymptotic approximation. Let w1 be some solution of
−Bw0 + Aw = w000 .
That means we want to construct a ”global expansion” of the form
(w0 + wh c̃0 ) + ε(w1 + wh c̃1 ).
To account for the boundary conditions a more careful study as in the first step for the local
expansions at x = 0 and at x = 1 is necessary. With ξ = x/ε we denote by v1 the exponential
decreasing solution of
d2 v
dv
+ B(0)
= A(0)v0 − B 0 (0)ξv00 .
(2.6)
2
dξ
dξ
4
Here v0 is the layer term at x = 0 in (2.4). Then v1 has the structure
· ¸
α
v1 = d3 1 exp(−λ̃1 ξ) + v1,p (ξ).
β1
v1,p is one decreasing solution of the inhomogeneous problem (2.6). That means: If λ∗1 < λ̃1 ,
then
¯ k ¯
¯ d v1 ¯
∗
¯
¯
¯ dξ k ¯ ≤ C exp(−λ1 ξ).
Putting the global expansion and the layer corrections together, our new asymptotic expansion reads
¶
· ¸
µ
· ¸
α2
α1
1
0
exp(λ̃2 (1 − x)/ε) + v2,p .
exp(−λ̃1 x/ε) + v1,p + d4
uas = uas + ε w1 + wh c̃1 + d3
β2
β1
The unknown constants d3 , d4 and the unknown vector c̃1 are determined by the homogeneous
boundary conditions.
The remainder R1 in the expansion
u = u1as + R1
satisfies the boundary value problem
LR1 = g1 ,
R1 |x=0 = R1 |x=1 = 0.
This time we have kg1 k0 ≤ Cε3/2 , which implies |R1 |2 ≤ C. Consequently, the first order
derivatives of R1 are pointwise bounded as well.
Summarizing we proved the following theorem.
Theorem 2.5
Suppose the data of the system (1.1) equipped with homogeneous boundary conditions is
sufficiently smooth. Let the conditions (V1 )-(V4 ) be satisfied. Then u can be decomposed as
u = S + E1 + E2 .
(2.7)
Here the smooth solution component S satisfies
kSk2 ≤ C,
(2.8)
while the layer terms are characterized by
(k)
|E1 | ≤ Cε−k exp(−λ∗1 x/ε),
(k)
|E2 | ≤ Cε−k exp(−λ∗2 (1 − x)/ε).
(2.9)
In general, both solution components do have layers at x = 0 and at x = 1. This means
we have full layer interaction due to the strong coupling of the equations.
Obviously our approach can be extended to higher order asymptotic approximations and
to some strongly coupled systems of higher dimension.
Remark 2.6
The case of two different parameters ε1 , ε2 in the system
− diag(ε1 , ε2 ) u00 − Bu0 + Au = f.
is more complicated. Now the asymptotic behaviour of the system depends on the eigenvalues
and eigenvectors of the matrix
·
¸
b11 /ε1 b12 /ε1
B̃ =
.
b21 /ε2 b22 /ε2
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Thus it can happen that the solution of the reduced problem also depends on ε1 , ε2 as the
following example shows. Consider
· ¸
¸
·
1
−1 1 0
00
− diag(ε1 , ε2 ) u −
u =
0 3
2
with homogeneous boundary conditions. Now the asymptotic approximation reads
·
¸
· ¸
· ¸
x/3
1
1
0
uas B
+ c̃0 + d1
exp(−3x/ε1 ) + d2
exp(−(1 − x)/ε2 ).
−2x/3
4
0
Computing the constants, we get for the reduced solution
2
u2,red = (1 − x)
3
as in the case ε1 = ε2 . However, for the first component we obtain
u1,red =
2
x
+
.
3 3 (1 + 3 ε1 /ε2 )
It is surprising that the solution of the reduced problem depends on the parameters ε1 , ε2 .
We shall study the two-parameter problem in a subsequent paper.
3 Computational results
Next we solve the system
−εu00 −
·
¸
·
¸
1 −1 12
x − x2
u0 =
1
5 12 −19
(3.1)
numerically. We use linear finite elements on a layer-adapted Shishkin mesh.
It can easily be checked that system (3.1) satisfies the assumptions (V1)–(V4), it has
therefore an unique solution (cf. lemma (2.1)). Furthermore this solution is known explicitly,
thus it is possible to calculate the error of the numerical solution in various norms as we
have done below.
From our asymptotic expansion we know that there are boundary layers of the form
exp(−|λ̃1 | x/ε) and exp(−|λ̃2 | (1 − x)/ε) at the left and right side of the domain, respectively.
Therefore we select the mesh transition points σ1 B min{1/3, 5/2 ε ln N/|λ̃1 |} and σ2 B
max{2/3, 1 − 5/2 ε ln N/|λ̃2 |} and use for each of the subdomains [0, σ1 ], [σ1 , σ2 ], [σ2 , 1] an
equidistant mesh with N/3 points. Note that for our example λ̃1 = 1 and λ̃2 = −5.
Theorem 2.5 allows us to use standard techniques as in [5, 7] to prove error estimates. We
get
ku − uN kε ≤ CN −1 ln N,
¡
¢2
kuI − uN kε ≤ C N −1 ln N ,
¡
¢2
ku − uN k0 ≤ C N −1 ln N ,
(3.2)
where uI denotes the nodal interpolant of the exact solution u and uN denotes the numerical
solution. Here the constant C is independent of ε.
In our computations we choose the perturbation parameter ε = 10−6 and ε = 10−8 . These
are sufficiently small for the solution to exhibit the typical boundary layers. We use meshes
with 10 to 105 interior nodes.
The computational results are presented in figure 1. It appears that the numerical errors
confirm the theoretical results stated in equation (3.2), especially the independence of the
convergenc rate of the parameter ε.
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Figure 1: Error of the numerical solutions measured in various norms
ε = 1e−06
0
10
−2
10
−4
10
−6
10
−8
10
||u − uN ||0
||u − uN ||ε
||uN − uI ||ε
ln(N )/N
(ln(N )/N )2
−10
10
1
2
10
10
3
10
4
10
5
10
ε = 1e−08
0
10
−2
10
−4
10
−6
10
−8
10
||u − uN ||0
||u − uN ||ε
||uN − uI ||ε
ln(N )/N
(ln(N )/N )2
−10
10
1
10
2
10
3
10
4
10
5
10
References
[1] L. R. Abrahamsson, H. B. Keller, H. O. Kreiss, Difference approximations for singular
perturbations of systems of ordinary differential equations, Numer. Math. 22, 367-391
(1974)
[2] T. Linß, Analysis of an upwind finite difference scheme for a system of coupled singularly
perturbed convection-diffusion equations, Computing 79, 23-32 (2007)
[3] T. Linß, Analysis of a system of singularly perturbed convection-diffusion equations
with strong coupling, SINUM 47, 1847-1862 (2009)
[4] T. Linß, M. Stynes, Numerical solution of systems of singularly perturbed differential
equations, Comput. Meth. Appl. Math. 9, 165-191 (2009)
[5] T. Linß, Layer adapted meshes for reaction-convection-diffusion problems (SpringerVerlag, Berlin Heidelberg, 2010)
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[6] E. O’Riordan, M. Stynes, Numerical analysis of a system of strongly coupled system
of two singularly perturbed convection-diffusion equations, Adv. Comput. Math., 30
101-121 (2009)
[7] H.-G. Roos, M. Stynes, L. Tobiska, Robust numerical methods for singularly perturbed
differential equations (Springer-Verlag, Berlin Heidelberg, 2008)
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