On the Coupled of NBEM and FEM for an Anisotropic Quasilinear Problem in Elongated Domains

Abstract

In this paper, based on the Kirchhoff transformation, the coupling of natural boundary element method and finite element method are discussed for solving exterior anisotropic quasilinear problems with elliptic artificial boundary. By the principle of the natural boundary reduction, we obtain natural integral equation on elliptic artificial boundaries, the coupled variational problem and its numerical method. Moreover, the convergence and error estimate of the approximate solutions are obtained. Finally, some numerical examples are presented to illuminate the feasibility of the method.

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Liu, B. and Du, Q. (2012) On the Coupled of NBEM and FEM for an Anisotropic Quasilinear Problem in Elongated Domains. American Journal of Computational Mathematics, 2, 143-155. doi: 10.4236/ajcm.2012.22020.

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 ofcan 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)

withand,.

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 operatorsatisfies

(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 andsatisfies (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 operatoris 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.

Figure 2. Example 4.1 with different N.

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.

Figure 4. Example 4.2 with different N.

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.

Figure 6. Example 4.3 with different N.

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 ofas 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.

Figure 8. Example 4.4 with different N.

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

Conflicts of Interest

The authors declare no conflicts of interest.

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