Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2007, Article ID 98538, 6 pages
doi:10.1155/2007/98538
Research Article
Homogenization of Elliptic Differential Equations
in One-Dimensional Spaces
G. Grammel
Received 16 August 2006; Revised 7 December 2006; Accepted 14 December 2006
Recommended by Vesa Mustonen
Linear elliptic differential equations with periodic coefficients in one-dimensional domains are considered. The approximation properties of the homogenized system are investigated. For H −1 -data, it turns out that the order of approximation is strongly related
to the decay of the Fourier coefficients of the L2 -functions involved.
Copyright © 2007 G. Grammel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
We consider the linear elliptic differential equation
d
d
a (x) u (x) = f (x)
dx
dx
(1.1)
on the interval x ∈ (0,1) with Dirichlet boundary conditions
u (0) = u (1) = 0.
(1.2)
Here, > 0 is a small parameter and a (x) = a(x/ ). Under natural suppositions on the
coefficient function a and the data f given below, the boundary problem given above
does possess a unique solution u ∈ H01 ((0,1); R).
Assumption 1.1. The space-dependent coefficient is of the form a (x) = a(x/ ), where
the mapping a : R → R is measurable, essentially bounded, and periodic. In particular,
there are constants 0 < α ≤ β < ∞ and Y > 0 such that 0 < α ≤ a(y) ≤ β < ∞ and a(y) =
a(y + Y ) for a.a. y ∈ R.
2
Abstract and Applied Analysis
Assumption 1.2. As for the data, we assume that f ∈ H −1 ((0,1); R) = (H01 ((0,1); R))∗ .
Accordingly, f is of the form f = f0 + (d/dx) f1 , with f0 , f1 ∈ L2 ((0,1); R), where d/dx
denotes the distributional derivative. Note that f0 , f1 are not uniquely determined by f .
For both numerical and theoretical considerations, it is quite advantageous to simplify the differential equation. Averaging leads to the so-called homogenized differential
equation
d
d
a0 u0 (x) = f (x),
dx
dx
(1.3)
where the constant coefficient is given by
1
1
=
a0 Y
Y
0
1
d y.
a(y)
(1.4)
Naturally, the corresponding Dirichlet boundary problem is uniquely solvable in
H01 ((0,1); R) as well. The relation between the original and the homogenized solutions
can be described as follows, see [1, Theorem 6.1].
Theorem 1.3. Let Assumptions 1.1, 1.2 be satisfied. Then one has the convergence relation
u → u0 weak in H01 ((0,1); R), as → 0.
There are several methods available to prove the homogenization result above. We
mention Tatar’s method displayed in [2] and the two-scale convergence method elaborated in [3]. Both methods are applicable to differential equations in higher dimensional
domains. It is well known, see [4], that the approximation is linear in > 0 for L2 -data,
but it seems that the order of approximation has not yet been investigated for H −1 -data.
In the present paper, we assume that the Fourier coefficients given by the H −1 -data have
a sufficiently fast decay and obtain approximation orders (with respect to the uniform
convergence) in dependence of the order of the decay of the Fourier coefficients. Here,
the usual linear order for L2 -data appears as a limit, as the H −1 -data approaches an L2 function. Note that the method of proof has nothing in common with the Fourier homogenization method, see, for instance, [5], since we do not use a Fourier analysis for
the rapidly varying coefficients.
2. Order of approximation
In order to obtain an order of approximation we have to suppose a sufficient fast decay
of the Fourier coefficients of f1 . Here we set
c0 :=
ck :=
for k = 1,2,3,....
1
0
1
cos(k2πx)
√
dx,
f1 (x)
2
0
f1 (x)dx,
sk :=
1
0
sin(k2πx)
√
dx,
f1 (x)
2
(2.1)
G. Grammel 3
Theorem 2.1. Let Assumptions 1.1, 1.2 be satisfied. Moreover, assume that the Fourier coefficients of f1 fulfill the estimation
ck = O k−γ ,
sk = O k−γ ,
as k −→ ∞,
(2.2)
for a γ > 1/2. Then the solutions u0 and u are continuous on [0,1] and, as → 0, one can
estimate
⎧ ⎪
γ−1/2 ,
⎪
⎪O ⎪
⎪
⎪
⎨
u − u0 = O log(),
∞
⎪
⎪
⎪
⎪
⎪
⎪
⎩O(),
1
3
<γ< ,
2
2
3
if γ = ,
2
3
if γ > .
2
(2.3)
d
ξ (x)
u (x) = .
dx
a (x)
(2.4)
if
Proof. The differential equation (1.1) is equivalent to
d
d
ξ (x) = f0 (x) +
f1 (x),
dx
dx
Hence, with an appropriate constant C ∈ R, we can write
ξ (x) =
x
0
x z
0
u (x) =
f0 (w)dw + f1 (x) + C ,
0
f0 (w)dw + f1 (z) + C
dz,
a (z)
(2.5)
for all > 0. The homogenized differential equation (1.3) is equivalent to
d
d
ξ0 (x) = f0 (x) +
f1 (x),
dx
dx
d
ξ (x)
u0 (x) = 0 .
dx
a0
(2.6)
Hence, with an appropriate constant C0 ∈ R, we can write
ξ0 (x) =
x
0
u0 (x) =
f0 (w)dw + f1 (x) + C0 ,
x z
0
0
f0 (w)dw + f1 (z) + C0
dz.
a0
(2.7)
Integrating by parts, we conclude that
u (x) − u0 (x)
=
=
=
x
0
x
0
x
0
C + F0 (z) + f1 (z)
dz −
a (z)
C − C0
dz +
a (z)
x
0
C0 + F0 (z) + f1 (z)
dz
a0
x
0
x
C0 g (z)dz +
C − C0
x
−
dz + C0 A
a (z)
x
0
0
x
F0 (z)g (z)dz +
f0 (z)A
z
0
f1 (z)g (z)dz
x
dz + F0 (x)A
x
+
0
f1 (z)g (z)dz,
(2.8)
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Abstract and Applied Analysis
where
g (z) :=
1
1
− ,
a (z) a0
F0 (z) :=
z
0
y
A(y) :=
f0 (w)dw,
0
1
1
−
dq.
a(q) a0
(2.9)
By the periodicity of a : R → R and A(0) = A(Y ) = 0, we easily obtain the estimation
1 1
−
.
max A(y) = max A(y) ≤ Y
y ∈R
0≤ y ≤Y
α β
(2.10)
Furthermore, we obviously have
max g (z) ≤
z∈R
1 1
− .
α β
(2.11)
The boundary condition 0 = u (1) = u0 (1) implies that
1
0
C − C0
1
+
dz = −C0 A
a (z)
1
0
f0 (z)A
z
dz − F0 (1)A
1
−
1
0
f1 (z)g (z)dz
(2.12)
and that
1
z
1 C − C0 ≤ β C0 A 1 + f0 (z)A
dz + F0 (1)A
0
1
.
+
f
(z)g
(z)dz
1
(2.13)
0
Overall, we obtain
u (x) − u0 (x) ≤
1
β
1 1 C0 + 2 f0 (z)dz
−
+ 1 Y
α
α β
0
x
β
+
f1 (z)g (z)dz
+ 1 max .
0≤x≤1
α
0
(2.14)
Let us note that the boundary condition u0 (1) = 0 gives us
C0 ≤ f0 2 + f1 2 ,
L
L
(2.15)
which yields
x
2
2
.
u − u0 ≤ β − α Y 3 f0 2 + f1 2 + β + 1 max f
(z)g
(z)dz
1
∞
L
L
2
αβ
α
0≤x≤1
0
(2.16)
In case that γ > 3/2, the distributional derivative (d/dx) f1 is square integrable, that is,
f ∈ L2 ((0,1); R) and we can assume that f1 = 0. By (2.16), we obtain the linear approximation. In case that 1/2 < γ ≤ 3/2, it only remains to obtain an estimation for the last
G. Grammel 5
summand in (2.16). To this end, we write with K ∈ N (to be specified later)
f1 (x) = c0 +
K ∞
cos(2πkx)
sin(2πkx)
cos(2πkx)
sin(2πkx)
√
√
√
√
ck
+
ck
.
+ sk
+ sk
2
k =1
2
2
k=K +1
2
(2.17)
As K → ∞, we can estimate
2
K d cos(2πk·)
sin(2πk·) √
√
ck
+ sk
dx
2
2
2
k =1
L
⎧
3−2γ ⎪
⎪
K ,
⎨O K
2
2
= (2π)2
kck + ksk
=
⎪
⎪
⎩O log K ,
k =1
K cos(2πk·)
sin(2πk·) √
√
c0 +
+ sk
ck
2
2
k =1
K
1 ck + sk =
≤ c0 + √
2 k =1
∞
⎧
3/2−γ ⎪
⎪
⎨O K
,
⎪
⎪
⎩O(1),
∞ 2
cos(2πk·)
sin(2πk·) √
√
ck
+ sk
2
2
2
k=K +1
=
∞
ck2 + s2k
k=K +1
=
⎧
1−2γ ⎪
⎪
⎨ O K
,
⎪
⎪
⎩O K−2 ,
1
3
<γ< ,
2
2
3
if γ = ,
2
if
1
3
<γ< ,
2
2
3
if γ = ,
2
if
(2.18)
L
1
3
<γ< ,
2
2
3
if γ = .
2
if
Integrating by parts, we can estimate
x
max 0≤x≤1
0
x
f1 (z)g (z)dz
≤ 0max
≤x ≤1 0
c0 +
K cos(2πkz)
sin(2πkz)
√
√
+ sk
ck
g (z)dz
2
2
k =1
∞
x cos(2πkz)
sin(2πkz)
√
√
+ max c
g
(z)dz
+
s
k
k
0≤x≤1 0
2
2
k=K +1
K d cos(2πk·)
sin(2πk·) √
√
≤
+ sk
ck
dx
2
2
k =1
A · L2
L2
K cos(2πk·)
sin(2πk·) √
√
+ sk
+ c0 +
ck
2
2
k =1
A · ∞
∞
∞ cos(2πk·)
sin(2πk·) √
√
g (·)L2 .
+ sk
+
ck
2
2
2
k=K +1
L
(2.19)
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Abstract and Applied Analysis
For 1/2 < γ < 3/2, the boundedness of A and g yields
x
3/2−γ 3/2−γ 1/2−γ + O K
+ O K
.
max f1 (z)g (z)dz
= O K
0≤x≤1
(2.20)
0
Choosing K ∈ N such that
K − 1 ≤
1
≤ K
(2.21)
yields the required estimation.
For γ = 3/2, the boundedness of A and g yields
x
max 0≤x≤1
0
f1 (z)g (z)dz
= O
log K
+ O(1) + O K−1 .
(2.22)
Choosing K ∈ N such that
K − 1 ≤
1 | log()|
≤ K
(2.23)
yields the required estimation.
Remark 2.2. The suppositions of Theorem 2.1 can be verified for f1 ∈ C([0,1]; R). Let
the modulus of continuity ω of f1 satisfy ω(δ) = O(δ γ ), as δ → 0, for a γ ∈ (1/2,1]. Then
the Fourier coefficients fulfill
ck = O k−γ ,
sk = O k−γ ,
as k −→ ∞,
(2.24)
see [6, Theorem 4.6].
References
[1] D. Cioranescu and P. Donato, An Introduction to Homogenization, vol. 17 of Oxford Lecture Series
in Mathematics and Its Applications, The Clarendon Press, Oxford University Press, New York,
NY, USA, 1999.
[2] L. Tartar, “Compensated compactness and applications to partial differential equations,” in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, R. J. Knops, Ed., vol. 39 of Research Notes in Mathematics, pp. 136–212, Pitman, Boston, Mass, USA, 1979.
[3] G. Allaire, “Homogenization and two-scale convergence,” SIAM Journal on Mathematical Analysis, vol. 23, no. 6, pp. 1482–1518, 1992.
[4] A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,
vol. 5 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 1978.
[5] C. Conca and F. Lund, “Fourier homogenization method and the propagation of acoustic waves
through a periodic vortex array,” SIAM Journal on Applied Mathematics, vol. 59, no. 5, pp. 1573–
1581, 1999.
[6] A. Zygmund, Trigonometric Series. Vol. I, II, Cambridge Mathematical Library, Cambridge University Press, Cambridge, UK, 3rd edition, 2002.
G. Grammel: Centre for Mathematics, Technical University of Munich, 3 Boltzmann Street,
85747 Garching, Germany
Email address: grammel@ma.tum.de