Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 828615, 16 pages

http://dx.doi.org/10.1155/2013/828615

Research Article

A Nonoverlapping Domain Decomposition Method for an Exterior Anisotropic Quasilinear Elliptic Problem in Elongated Domains

Baoqing Liu and Qikui Du

Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu 210023, China

Received 2 September 2012; Accepted 17 December 2012

Academic Editor: Yong-Kui Chang

Copyright © 2013 Baoqing Liu and Qikui Du. 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.

Abstract.

Based on the Kirchhoff transformation, a nonoverlapping domain decomposition method is discussed for solving exterior anisotropic quasilinear problems with elliptic artificial boundary. By the principle of the natural boundary reduction, we obtain the natural integral equation for the anisotropic quasilinear problem on elliptic artificial boundaries and construct the algorithm and analyze its convergence. Moreover, we give the existence and uniqueness result for the original problem. Finally, some numerical examples are presented to illustrate the feasibility of the method.

1. Introduction

When solving a problem modelled by a linear or nonlinear partial differential equation in the bounded or unbounded domain, domain decomposition methods are one of the most efficient techniques. one can refer to [1–4] and references therein for more details. Based on natural boundary reduction [5, 6], the overlapping and nonoverlapping domain decomposition methods can be viewed as effective ways to solve the problems in the unbounded domains. These techniques have been used to solve many linear problems [6–8], and they have also been generalized to linear or nonlinear wave problems [5, 9, 10]. In this paper, we consider a nonoverlapping domain decomposition method for an exterior anisotropic quasilinear elliptic problem with elliptical artificial boundary. By the Kirchhoff transformation, we will discuss some exterior anisotropic quasilinear elliptic problems [11–15] by virtue of the nonoverlapping domain decomposition method.

Let be an elongated, bounded, and simple connected domain in with sufficiently smooth boundary , . We consider the numerical solution to the exterior anisotropic quasilinear problem

with or , , , and are the given functions which will be ranked as below.

Problem (1) has many physical applications in, for example, the field of heat transfer, where is the thermal conductivity of the medium and is the temperature field; the field of compressible flow, where is the density and is the velocity potential. Problem (1) can also describe a temperature distribution in large transformers whose magnetic cores (consisting of iron tins) are nonlinear anisotropic media where is the heat conductivity. One can also refer to [11–14, 16] for more details.

Following [11, 13], suppose that the given function satisfies

Where the two constants . In the following, we suppose that the function has compact support, that is, there exists a constant , such that

We also assume that

Now, we introduce an elliptical artificial boundary enclosing such that

Then, is divided into two nonoverlapping subdomains and (see Figure 1), where denotes the bounded domain between and , and refers to the unbounded domain outside . The original problem (1) is decomposed into two subproblems in domains and with . We have the nonoverlapping domain decomposition algorithm.

Figure 1: The illustration of domains and .

Step 1. Choose an initial value , and put .

Step 2. Solve a Dirichlet boundary value problem in the exterior domain :

Step 3. Solve a mixed boundary value problem in the interior domain :

Step 4. Update the boundary value ,

Step 5. Put , turn to Step 2.

The relaxation factor is a suitably chosen real number. Notice that, in Step 3, we solve problem (8) by the standard finite element method and only need the normal derivative of the solution to problem (7) in Step 2. So, we need not to solve (7) directly. Based on the Kirchhoff transformation, the natural integral equation for the quasilinear problem can be obtained by the natural boundary reduction [6, 17, 18]. Particularly, when which is independent of and , [6, 8, 19] have discussed the corresponding problems by this technique. Now, we introduce the so-called Kirchhoff transformation [20]

Then, we have

From (7), we have that satisfies the following problem

The rest of the paper is organized as follows. In Section 2, we obtain the natural integral equation for the quasilinear problem in the elliptical unbounded domain. In Section 3, we state the nonoverlapping domain decomposition method and discuss the convergence of the algorithm. We also show the existence and uniqueness of the original problem. At last, in Section 4, we present some numerical examples to illustrate 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 will 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

with , , and . Following from [18], we have the following.

Theorem 1. The transformation between elliptic coordinates and Cartesian coordinates in (14) possesses the following property.

(1)The Jacobian determinant of (14) is

if and only if .(2)For , the following holds

(3)For the exterior domain , we have

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

Assume that is the solution to problem (13) and the value is given, that is,

Then, based on the natural boundary reduction, there are the Poisson integral formulas

or

and the natural integral equation

or

the definition of can be found in the following. The Poisson integral formulas (20)-(21) and the natural integral equation (22)-(23) can also be expressed in the Fourier series forms

with , , .

From (12), we obtain

Combining (11), (25), and (26), we get the exact artificial boundary condition of on

with , , .

2.2. Natural Integral Equation for

Now, we assume that can be expressed in the form: , with . We also assume that is the solution to problem (13) and the value is given, namely,

Let and . Then, the boundary is changed by the elliptic boundary ; the unit exterior normal vector on is

By the above transformation, problem (13) changes into

This is the right problem we talked in Section 2.1. Similar with (14), we let

with

Then, just the same as the problem discussed in Section 2.1, we have the natural integral equation on

where is the unit exterior normal vector on . From (13), we obtain

Combining (11), (34), and (35), we obtain the exact artificial boundary condition of on

3. Variational Problem and Convergence Analysis of the Algorithm

3.1. The Equivalent Variational Problem

Now, we consider problem (8). We will use denoting the standard Sobolev spaces. and refer to the corresponding norms and seminorms. Especially, we define , , and . Let us introduce the space

and the corresponding norms

The boundary value problem (8) is equivalent to the following variational problem.

with

where follows from Green’s formula, (27) with , and (36) with .

And,

3.2. Convergence Analysis of the Method in Continuous Case

From (10)-(11), the original problem (1) can be changed to

Then, we let and . Problem (43) becomes

where and are the corresponding changes from and , respectively. Let be extended to , , . Then, problem (44) is equivalent to

Since it is difficult to estimate the convergence rate for a general unbounded domain , we here let be an exterior domain of an ellipse , with , and is taken as stated in Section 1. We introduce the following conclusion first.

Lemma 2. If is the solution of

where is the elliptical ring domain between and ,

then there exist a unique and

Proof. The result can be obtained directly from (46) by the separation of variables.

Theorem 3. If , then the nonoverlapping domain decomposition method (7)–(9) is convergent.

Proof. We assume that the exact solution to problem (1) is , and we let , , and . Then, following (7)–(8), we let and . From (24), we assume that . By the natural integral equation (27), we have

So, satisfies

By Lemma 2, one obtains

with . From (51), confines on can be expressed as

Then, we have

Let , then and

Assuming that , then .

If , , then

or equally

By the trace theorem, we have

From (54), one also has

with . Assuming that , then .

For , , the convergence result can be obtained similarly with (55)–(57). Therefore, for , the nonoverlapping domain decomposition method is convergent.

3.3. Discrete Nonoverlapping Alternating Algorithm and Convergence Analysis

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

where is a (curved) triangle and is the maximal diameter of triangles. Let

Then, the approximate problem of (39) can be written as follows

with

In practice, the sum of (63) is truncated to a finite number of terms . By the hypothesis of , it is not difficult to know the following result.

Lemma 4. There exists a constant which has a different meaning in a different place and is related to and , such that

Proof. From (3) and (40), one can obtain that

For the natural integral equation, one can obtain the following result

For this purpose, for , we assume that

with , . Then, we have

Combining property (3), Cauchy’s inequality, and the trace theorem, we have

Next, we show that , for any . For any given , let us consider the following auxiliary problem in :

From the analysis in Section 2.1, we know that the solution to problem (70) satisfies

Multiplying (70) by and integrating over , we have

Letting , we can get the desired result.

We are now in the position to show the existence and uniqueness result for this type of problem (1). We begin with the following estimate.

Lemma 5. Any solution to problem (39) satisfies

with a positive constant.

Proof. Taking equal to in (39), by Lemma 4, one has

Then, the desired result follows from Cauchy-Schwarz and Poincaré-Friedrichs inequalities.

Theorem 6. Problem (39) is uniquely solvable.

Proof. Since the space is separable (indeed, it is a closed subspace of the space which is separable), there exist increasing sequences of finite-dimensional Hilbert subspaces of such that

We define a mapping by

where stands for the scalar product on . Since the function is bounded and the trace of the function in on belongs to , hence at least to from the Sobolev embedding, each mapping is well defined and continuous on . What’s more, by Lemmas 4 and 5, we obtain

So, the right-hand side is nonnegative on the circle of radius which is defined by

Applying Brouwer’s fixed point theorem [21] yields the existence of in , with norm less than , such that

Since the sequence is bounded by in , there exists a subsequence which converges weakly to in . Using the compactness of the imbedding of into , we obtain that is a solution to problem (39).

Now, we show the uniqueness of the solution. Let and be two solutions to this problem. Then, taking equal to and in (39), respectively, and combining with Lemma 4, one obtains

Since , combining (80) with (81), we deduce the desired result.

From the discrete problem (61), similar to the discussion for the linear problem in [6, 8], we can get a system of algebraic equations for our quasilinear problem with the following form:

where is a vector whose components are function values at nodes on , and is a vector whose components are function values at interior nodes of . The matrix is the stiffness matrix obtained from finite element in , while follows from the natural boundary element method on .

Problem (82) can also be rewritten as follows:

Then, we have the iterative algorithm

with

By condition (3), one obtains that is a positive definite matrix, so we know that exists. Now, we let be the discrete analogue of the Steklov-Poincaré operator on , with and . Then, similar to the proof of Theorems 7.6 and  7.7 in [6], we have that the alternating algorithm (84)-(85) is equivalent to the preconditioned Richardson iteration:

And we also have the following convergence result.

Theorem 7. If , then the discrete nonoverlapping alternating method (84)-(85) is convergent, and both the convergence rate and the condition number of are independent of the mesh size .

4. Numerical Examples

In this section, we will give some examples to confirm our theoretical results. In the following, we choose the finite element space as given in (60). For simplicity, we let

And we let denote the maximal error between the iteration and , that is, and simulate the convergence rate.

We give the following four examples. Examples 8 and 11 show the relationship between meshes and convergence rate for the cases and , respectively. Example 9 focuses on the effect of the relaxation factor to the convergence rate. And Example 10 wants to show the relationship between the coefficients and the convergence rate.

Example 8. We assume the exterior domain with elliptical boundary , . Now, we consider the problem

when , , and .

The exact solution of Example 8 is . For different meshes and , we have the following results ranked in Tables 1 and 2.

Table 1: The relationship between meshes and convergence rate (, , and ).

)
Error
Iteration number

0
1
2
3
4
5
6
9

(2,8)

—

—
—
1.7980
1.8620
1.8077
1.7220
1.6296
1.3992

(4,16)

—

—
—
2.2192
2.3452
2.2467
2.1233
1.9925
1.5865

Table 2: The relationship between meshes and convergence rate (, , and ).

, )
Error

Iteration number

0
1
2
3
4
5
6
9

(2,8)

—

—
—
1.6309
1.6194
1.5564
1.4880
1.4248
1.2857

(4,16)

—

—
—
1.8516
1.8787
1.7868
1.6828
1.5860
1.3728

Example 9. Similar with Example 8, and are replaced by , , and , respectively.

The exact solution of Example 9 is . The numerical results are given in Table 3 and Figure 2(a).

Table 3: The relationship between and convergence rate (, , and ).

0
1
2
3
4
5
6
9

0.18

—

—
—

0.38

—

—
—

0.50

—

—
—

0.58

—

0.65

—

—
—

(a)

(b)

(c)

Figure 2: (a)   The relationship between and convergence rate (, and ). (b) The relationship between and convergence rate (, , and ). (c) The relationship between meshes and convergence rate (, ).

Example 10. We assume the exterior domain with elliptical boundary , . Now, we consider the problem

when , and

The exact solution of Example 10 is . The numerical results are given in Table 4 and Figure 2(b).

Table 4: The relationship between and convergence rate (, , , and ).

0
1
2
3
4
5
6
9

0.75

—

—
—

0.5

—

—
—

0.05

—

—
—

0.005

—

—
—

0.0005

—

—
—

Example 11. Similar with Example 10, , , and are replaced by , , and , respectively. And we take

The exact solution of Example 11 is . The numerical results are given in Tables 5 and 6 and Figure 2(c).

From the numerical results, one obtains that the numerical errors can be affected by the choice of relaxation factor , the coefficient , and the location of the artificial boundary. In Tables 1, 2, 5, and 6 and Figure 2(c), the relationship between the meshes and the convergence rate is presented. We obtain that the convergence rate is independent of the finite element mesh size. In Table 3 and Figure 2(c), the convergence rates for different relaxation factor are compared. The results indicate that the choice of the relaxation factor is very important for the performance of the nonoverlapping alternating method. On the other hand, the convergence rate is not sensitive to the relaxation factor between the interval . What’s more, as it is shown in Table 4 and Figure 2(c), for the anisotropic problems, the choice of can also affect the convergence rate.

Table 5: The relationship between meshes and convergence rate (, , and ).

, )
Error
Iteration number

0
1
2
3
4
5
6
9

(2,8)

—
—
2.7176
2.2557
2.0684
1.9004
1.7452
1.3973

(4,16)

—

—
—
3.3154
2.5958
2.3738
2.1772
1.9774
1.5007

Table 6: The relationship between meshes and convergence rate (, , and ).

(, )
Error
Iteration number

0
1
2
3
4
5
6
9

(2,8)

—
—
3.0780
2.3746
2.1829
2.0122
1.8465
1.3787

(4,16)

—
—
4.3449
3.1069
2.5771
2.2790
2.0543
1.4481

Acknowlegments

This work is supported by the National Natural Science Foundation of China, Contact/Grant no. 11071109 and the Foundation for Innovative Program of Jiangsu Province, Contact/Grant no. CXZZ12_0383 and CXZZ11_0870.

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