A Finite Element Algorithm And Its
Convergence Analysis
for Elliptic Problems with Small Periodic
Structure
WEN-MING HE
Department of Mathematics, Wenzhou University, Wenzhou,Zhejiang,P. R.
China,325035
JUN-ZHI CUI
Institute of Computational Mathematics and Scientific/Engineering
Computing,CAS, P. O. Box 2719, Beijing,P. R. China, 100080
Email: [email protected]
PHONE 13758472823
Abstract In this paper,we propose a finite element algorithm for elliptic problems
with small periodic structure and present its convergence analysis,whose key is to compute the
classical boundary corrector of those equations from different meshes exploiting need for
different levels of resolution.Numerical experiments are included to illustrate the competitive
behavior of the proposed finite element method.
Key words The method of multiscale asymptotic expansions,
Finite element Algorithm, Second order elliptic equations with rapidly
oscillating coefficients.
1
Introduction
The method of multiscale asymptotic expansions which is thoroughly described in
numerous sources(see e.g. [1][2][3][4]) can be used to solve all kinds of second
order elliptic problems with rapidly oscillating coefficients very effectively,
for it couples the macroscopic and microscopic scales together.
For the following elliptic model problem
1




ij x u ( x )
(a (  )
)  f ( x), x  ,
 L u ( x) 
xi
x j

 
x  .(1.1)
u ( x)  g ( x),
The key of the method of multiscale asymptotic expansions is to obtain
^
the 1-order approximation u ( x) of u  (x) .Oleinik etc(see [1])noticed that
^
u  ( x)  u ( x) can be splitted into
^
u  ( x)  u ( x)  W  ( x)    ( x) ,
where W  (x) and   (x) satisfy
|| W  || H 1 (  )  C ,
||  || H 1 (  )  C 1 / 2 ,
and then obtain
^
|| u   u || H 1 (  )  C 1 / 2 .
Notation
  (x) is called the classical boundary corrector of u  (x) .
We want to say that there are many kinds of multiscale numerical method
To solve (1.1) and similar problems on basis of the method of multiscale
asymptotic expansions.For example,J.Z. Cui and L.Q Cao etc(see [5][6] )
proposed a method of multiscale Asymptotic expansions and its corresponding
finite element method;Yi-zhou Hou,Zhiming,Chen etc(see [7][8] ) proposed a
multiscale finite element Method; Weinan E, Pingbing Ming and Pingwen
Zhang(see [9][10]) presented a heterogeneous multiscale finite element method.
We notice that |   ( x) | is very large when dist ( x, ) is very
Small and there exists C such that
^
(u   u )
C.
  dist ( x, )
L2 ( )
In this paper,we present a finite element method to compute   (x) from
different meshes exploiting need for different levels of resolution.On basis of
it,we have a finite element algorithm to compute u  (x) .
2
A finite element algorithm for the problem (1.1) based on the
method of multiscale asymptotic expansions
2
In this section,we take two following steps to compute u  (x) .
(i)
Using multiscale finite element algorithm,we obtain a numerical
^
approximation of the 1-order approximation u ( x) of u  (x) .
(ii)
Using finite element algorithm to compute the classical
boundary corrector   (x) of u  (x) whose key is using the
following way to get a partition of  .
Firstly, assume that m N such that
2m 1  max dist ( x, )  2m 
x
we decompose  into
   i
0i  m
with
 i  {x   | dist ( x, )   },

i 1
i
 i  {x   | 2   dist ( x, )  2  },
i  0,
if
if
1  i  n,
Then, define a regular triangular partition  h of  as follows:
Assume that mesh e satisfies e  i  i 1 (0  i  m  1) ,
then there exists a constant  , independent of i ,such that
2i / 2   S e  2i / 2 1
where S e denotes the area of e .
On basif of (i) and (ii),we get the following finite element approximation
^ h
u
h0 , h1
Of
u  (x) as follows
^ h
u
h0 , h1
^ h1
^  ,h
( x) = u h0 ( x) + h
0 , h1
( x)
The following Lemma 2.1 gives an error estimate between
^
the 1-order approximation u ( x) of u  (x) and its finite element approximation
^ h1
u h0 ( x) and Lemma 2.2 gives an error estimate between
the classical boundary corrector   (x) of u  (x)
3
^  ,h
and its finite element approximation

h0 , h1
( x) .
Lemma 2.1 There exists C such that
^
^ h1
(u  u h )
 C (h0  h1   ) ,
0
  dist ( x, )
L2 (  )
and
Lemma 2.2 There exists C such that
^  ,h
(  h

0 , h1
)
 C (h0  h1    h).
  dist ( x, )
L2 (  )
Together Lemma 2.1 and Lemma 2.2,we have the main result of this paper.
Theorem 2.1 There exists C such that
^ h
(u   u h
0 , h1
)
 C (h0  h1    h)
  dist ( x, )
L2 (  )

^ h
Notation Theorem 2.1 show that (u  u h
0 , h1
Even if
3
)( x) is very small
dist ( x, ) is very small.
Numerical Experiments
Considering it is extremely difficult to construct a test problem with sufficient
generality and exact solution u  (x) ,we replace u  (x) by employing u  ,h which
Is obtained on basis of a quasi-uniform partition  h of  with grid size h .
4
Table 3.1 Numerical Experiments for the
finite element algorithm

h0
h1
h

^ h
(u  u h
0 , h1
)
  dist ( x, )
L2 (  )
1
24
0.2=0.2
0.2
0.2
0.089
1
24
0.1
0.1
0.1
0.047
1
24
0.05
0.05
0.05
0.029
Table 3.1 show that Theorem 2.1 is right and the finite element
Algorithm is very competitive.
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5
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