On the Coupled of NBEM and FEM for an Anisotropic Quasilinear Problem in Elongated Domains ()
1. Introduction
Based on the Green’s function and Green’s formula, natural boundary element method (NBEM) reduces the boundary value problem of partial differential equation into a hypersingular integral equation on the boundary, and then solves the latter numerically [1,2]. It has advantages over the usual boundary reduction methods: such as the diminution of the number of space dimensions by 1, the conservation of energy functional, the preservation of self-adjointness and coerciveness. But it also has evident limitations, it’s difficult to obtain Green’s functions for solving problem in general domains. Therefore, the coupling of NBEM which is also called artificial boundary condition [3,4] or DtN method [5,6] and finite element method (FEM) [2] is useful and necessary for general cases.
The standard procedure of the coupling method can be described as follows. We introduce an artificial boundary to divide the original domain into two subregions, a bounded inner region and an unbounded one with a special boundary, such as circle, ellipse, and spherical surface, on which the boundary element method and finite element method are used respectively. This technique has been used to solve many linear problems [1,2,4-6] and it has also been successfully generalized to solve nonlinear boundary value problems [7-9] or quasilinear problems [3,10,11]. The problems were discussed in [3,10,11] take circle as artificial boundary, but for the problems with elongated domains, an elliptic boundary that leads to a smaller computational domain is obviously better than the circle one. The purpose of the paper is to study the coupling of NBEM and FEM to solve the anisotropic quasilinear problems with an elliptic artificial boundary.
Let 
be a elongated, bounded and simple connected domain in 
with sufficiently smooth boundary
.
. We consider the numerical solution of the exterior anisotropic quasilinear problem
(1.1)
With 
or
, 
, 
and 
are given functions which will be ranked as below. Following [3,12], suppose that the given function 
satisfies
(1.2)
where two positive constants
, and
(1.3)
with a constant
. We also assume that
, 
are continuous. In the following, we suppose that the function 
has compact support, i.e., there exists a constant
, such that
(1.4)
We also assume that
(1.5)
Now, we introduce an elliptic artificial boundary
divide 
into two regions, a bounded domain 
and an unbounded domain 
with elliptic artificial boundary. Then the problem (1.1) can be rewritten in the coupled form:
(1.6)
(1.7)
(1.8)
where 
is the unit exterior normal vector on
. Particularly, when 
which is independent of 
and
, [13-15] have obtained the natural integral equation. We introduce the so-called Kirichhoff transformation [16]
(1.9)
then we have
(1.10)
and
(1.11)
From equation (1.7) we have that 
satisfies the following problem
(1.12)
The rest of the paper is organized as follows. In section 2, we obtain the natural integral equation for elliptic unbounded domain cases. In section 3, we give the equivalent variational problems and the finite element approximations. The reduced problem’s well-posedness, the convergence results and error estimate are also discussed. At last, in section 4, we present some numerical examples to illuminate the efficiency and feasibility of our method.
2. Natural Boundary Reduction
In this section, by virtue of the Poisson integral formula and natural integral equation for the linear problem, we shall obtain the corresponding results for the quasilinear problem in
. For this purpose, we need to discuss some properties between elliptic coordinates 
and Cartesian coordinates 
first. The relationship between the two coordinates can be expressed as below
(2.1)
where
, 
,
. Following from [15], we have
Theorem 2.1 The transformation between elliptic coordinates and Cartesian coordinates (2.1) possesses the following property.
1) The Jacobi determinant of equation (2.1) is
(2.2)
if and only if
;
2)
(2.3)
for
;
3) For the exterior domain 
(2.4)
where 
refers to the unit exterior normal vector on 
(regarded as the inner boundary of
).
Proof The conclusions 1 and 2 can be obtained by direct computation. And 3 follows from the property
2.1. Natural Integral Equation for α = β = 1
Assume that 
is the solution of the problem (1.12), and the value 
is given, namely
Then based on the natural boundary reduction, there are the Poisson integral formulas
(2.5)
or
(2.6)
And the natural integral equation
(2.7)
or
(2.8)
the definition of
can be found in the following. The Poisson integral formulas (2.5) and (2.6) and the natural integral equations (2.7) and (2.8) can also be expressed in the Fourier series forms
(2.9)
(2.10)
where
, 
and
From (1.10), we obtain
(2.11)
Combining (1.9), (2.10) and (2.11), we get the exact artificial boundary condition of 
on
,
(2.12)
where
, 
,
. Then by (1.6)-(1.8) and (2.12), the original problem with 
confines in 
can be defined as follows
(2.13)
Therefore, the solution of problem (2.13) is the solution of the problem (1.1) with 
confining in the bounded domain
.
2.2. Natural Integral Equation for β > α > 0
Now we assume that 
can be expressed in the form:
, with
. We also assume that 
is the solution of the problem
(1.12), and the value 
is given, namely
Let
, 
, then the boundary 
is changed by the elliptic boundary
the unit exterior normal vector on 
is
By the above transformation, the problem (1.12) changes into
(2.14)
This is the right problem we talked in section 2.1. Similar with equation (2.1), we let
where
Then just the same as the problem discussed in Section 2.1, we have the natural integral equation on 
(2.15)
where 
is the unit exterior normal vector on
. From (1.11), we obtain
(2.16)
Combining (1.9), (2.15) and (2.16), we obtain the exact artificial boundary condition of 
on
,
(2.17)
Then by (1.6)-(1.8) and (2.17), the original problem with 
confines in 
can be defined as follows
(2.18)
Therefore, the solution of problem (2.18) is the solution of the problem (1.1) with 
confining in the bounded domain
.
3. Variational Problem and Finite Element Approximation
3.1. The Equivalent Variational Problems
Now we consider the problems (2.13) and (2.18). We shall use 
denoting the standard Sobolev spaces, 
and 
referring to the corresponding norms and semi-norms. Especially, we define
, 
and
. Let us introduce the space
(3.1)
and the corresponding norms
The boundary value problems (2.13) and (2.18) are equivalent to the following variational problem
(3.2)
where
(3.3)
(3.4)
where 
is gotten from Green’s formula, (2.7) and (2.8) with 
and (2.17) with
.
(3.5)
For any real number
, we let
(3.6)
with
and
,
.
Lemma 3.1 There exists a constant 
which has different meaning in different place and is related to 
and
, such that
In practice, we need to truncate the series in (2.12) and (2.17) for some nonnegative integer
, that is
(3.7)
with
(3.8)
when
, and
(3.9)
when
. So we only use the summation of the first 
terms in (2.13) and (2.18). We will consider the following approximate problem
(3.10)
(3.11)
Both (3.10) and (3.11) are equivalent to the following variational problem
(3.12)
where
(3.13)
Similar with Lemma 3.1, we have
Lemma 3.2 There exists a constant 
which has different meaning in different place, such that
3.2. Finite Element Approximation
Divide the arc 
into 
parts and take a finite element subdivision in 
such that their nodes on 
are coincident. That is, we make a regular and quasiuniform triangulation 
on
, such that
(3.14)
with 
is a (curved) triangle; 
the maximum side of the triangles. Let
(3.15)
Then the approximate problem of (3.12) can be written as
(3.16)
Some existence and uniqueness results for this type of problem are given in [12,17,18] under some conditions on the coefficients
, so by the constraint conditions
(1.2) and (1.3) we have Lemma 3.3 Problems (3.2), (3.12) and (3.16) have unique solvability.
3.2.1. Convergence Theorems
In this section, we obtain the convergence result of the problems discussed above. We let 
and 
be the solution of problems (3.2), (3.12), (3.16) respectively. We also assume that
(3.17)
And we require that 
is a family of finitedimensional subspaces of
, which satisfies for any
(3.18)
(3.19)
where 
is independent of
.
The continuous piecewise polynomial spaces, such as (3.15), satisfy the condition (3.17). And if we let
, where 
is the interpolation operator, then by (3.19), we have
And we can also obtain the following result.
Lemma 3.4
.
Proof From the (1.2), (3.12) and Lemma 3.2, we have
For
, we assume that
with
and
.
Then we have
From
, we obtain that 
is bounded in
. Therefore, there exists a subsequence 
such that
. Then similar with the proof of Lemma 3.4 of [3], we obtain
By the above lemmas, we get the following convergence result.
Theorem 3.1 Let
, and the assumptions (3.17)-( 3.19) be satisfied, then we have
(3.20)
3.2.2. Error Analysis
In the following, we shall get error estimates for the approximate solution obtained from a FEM-NBEM discrete scheme in the cases
. We assume that the solution 
of problem (1.1) satisfies
For simplicity let us define the following notation
Then (3.2), (3.12), (3.16) can be replaced by the corresponding simple forms respectively.
Now we introduce the bilinear form 
and 
defined by
Let 
be the dual space of
. By (1.2) and continuity of
, we obtain that 
is bounded in
. Then there exists an operator 
such that
(3.21)
Similar with the proof of [10], we have the lemma as follows
Lemma 3.5 The bilinear form 
defined by 
satisfies the following inequality
(3.22)
where 
is a sufficient large constant and
.
We assume that
(3.23)
Let 
be the canonical injection. Since 
is compactly embedded in
, we have that the operator 
defined by 
is also compact. By (3.21) and (3.23) and 
satisfies the property of
, we obtain that 
is an isomorphism.
By the conditions (3.2), (3.22), (3.23) and Theorem 10.1.2 of [20], one can get that there exists
, such that the following inequality is satisfied
(3.24)
for some constant 
independent of
.
We define the Galerkin projection with respect to
, 
Then the operator
satisfies
(3.25)
We define the set
Lemma 3.6 
is a solution of (3.14) if and only if the following equation
holds, where
with
,
.
Proof. Let
, then by
and
We can get the desired result.
Let
. Then following [10,11], we have
Lemma 3.7 There exists a positive constant C independent of h, such that
We also have the following result.
Lemma 3.8 
Proof For any
, we only need to show that
.
Since 
is regular and quasi-uniform, referring to [19], we obtain the following inverse inequality
Combining the above inequalities with the definition of 
and (3.26), we obtain
By the definition of
, we get the desired result.
Theorem 3.2 Assume 
be the solution of (1), with
, 
, and we also assume that 
and
satisfies (3.23). With sufficiently small
, the finite element equation (3.16) has the approximate solution 
such that
where C is a constant independent of h and N.
Proof Firstly, for any
, we have
Then by (3.12), we have
Let
, we have
From (3.2), (3.22), (3.23) and [20], we obtain
(3.26)
We denote a nonlinear mapping
, which satisfies that for any given
, 
is the unique solution of
(3.27)
Therefore, we have
Combining the above equation with (3.25), we obtain the operator
is continuous, i.e.,
Next, we assume that
, then by Lemma 3.8, we have that
. By the definition of
, (3.27) can be rewritten as
Then, from (3.24), Lemma 3.6 and Lemma 3.7, we have
This implies that
. And since 
is also continuous, following from Brouwer’s fixed theorem, one can obtain that there exists
, such that
. From
Lemma 3.6, we deduce that 
is the solution of (3.16). What’s more, by (3.25) and the fact
, we obtain
(3.28)
Combining (3.26) with (3.28), one can obtain
This completes the proof.
4. Numerical Examples
In this section, we shall give some examples to confirm our theoretical results. In the following, we choose the finite element space as given in (3.16). For simplicity, we let
Example 4.1 We assume the exterior domain 
with elliptical boundary
Now we consider the problem
(4.1)
when
, 
and
.
The exact solution of Example 4.1 is
.
The numerical results are given in Figures 1 and 2 and Table 1.
Example 4.2 Similar with Example 4.1, 
and 
are replaced by
and 
respectively.
Figure 1. Example 4.1 with N = 16, µ1 = 1.7.
Table 1. The errors with N = 16 for Example 4.1.
The exact solution of Example 4.2 is
.
The numerical results are given in Figure 3, Figure 4 and Table 2.
Figure 3. Example 4.2 with N = 6, µ1 = 1.0.
Table 2. The errors with N = 6 for Example 4.2.
Example 4.3 We assume the exterior domain 
with elliptical boundary
Now we consider the problem
(4.2)
when
, 
and
The exact solution of Example 4.3 is
.
The numerical results are given in Figures 5 and 6 and Table 3.
Example 4.4 Similar with Example 4.3, 
and 
are replaced by
Figure 5. Example 4.3 with N = 10, ε = 0.005.
and 
respectively. And we take
The exact solution of Example 4.3 is
.
The numerical results are given in Figures 7 and 8 and Table 4.
From the numerical results, one obtains that the numerical errors can be affected by the order of artificial boundary condition, the mesh of the domain and the location of the artificial boundary, and it can be reduced by increasing the order of the artificial boundary condition and refining the mesh. What’s more, the convergence rate of anisotropic problems can also be affected by the choice of
as it is shown in Tables 3 and 4. The numerical results are in agreement with the error analysis we obtain and show the efficiency of the coupling method.
Table 3. The errors with N = 10 for Example 4.3.
Figure 7. Example 4.4 with N = 5, ε = 0.05.
Table 4. The errors with N = 10 for Example 4.4.
5. Acknowledgements
This research was partly supported by the National Natural Science Foundation of China (contact/grant number: 10871100,11071109). The computations in this paper have been carried out in the Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems. The authors express their thanks to them.
NOTES