Journal of Applied Mathematics

Volume 2013 (2013), Article ID 736834, 9 pages

http://dx.doi.org/10.1155/2013/736834

Research Article

The Modified Trapezoidal Rule for Computing Hypersingular Integral on Interval

Jin Li1,2 and Xiuzhen Li¹

¹School of Science, Shandong Jianzhu University, Jinan 25010, China

²School of Mathematics, Shandong University, Jinan 250100, China

Received 3 August 2013; Revised 12 October 2013; Accepted 12 October 2013

Academic Editor: Kazutake Komori

Copyright © 2013 Jin Li and Xiuzhen Li. 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.

The modified trapezoidal rule for the computation of hypersingular integrals in boundary element methods is discussed. When the special function of the error functional equals zero, the convergence rate is one order higher than the general case. A new quadrature rule is presented and the asymptotic expansion of error function is obtained. Based on the error expansion, not only do we obtain a high order of accuracy, but also a posteriori error estimate is conveniently derived. Some numerical results are also reported to confirm the theoretical results and show the efficiency of the algorithms.

1. Introduction

Consider the following integral:

where denotes a Hadamard finite-part integral ( is called hypersingular integral and is called supersingular integral) and is the singular point. The formulation of these classes of boundary value problems in terms of hypersingular integral equations has drawn lots of interest. Many scientific and engineering problems, such as acoustics, electromagnetic scattering, and fracture mechanics, can be reduced to boundary integral equations with hypersingular kernels. There exist several definitions, equivalent mathematically, for such kind of integrals in some literatures.

We mention the following one:

Accurate calculation of boundary element methods (BEM) arising in boundary integral equations has been a subject of intensive research in recent years. The hypersingular integrals have certain properties different from regular and weak singular integrals. One of the major problems arising from boundary element method, for solving such integral equations, is how to evaluate the hypersingular integrals on the interval or on the circle efficiently.

Hypersingular integral must be considered in Hadamard finite-part sense. Numerous works [1–18] have been devoted towards developing efficient quadrature formulas. In 1983, the series expansion of hypersingular integral kernel on circle was firstly suggested by Yu [19]. He solved the harmonic and biharmonic natural boundary integral equations successfully. The Newton-Cotes methods to compute the hypersingular integral on interval were firstly studied by Linz [20] with generalized trapezoidal and Simpson rules which fail altogether when the singular point is close to a mesh point. In order to make the mesh be selected in such a way that falls near the center of a subinterval, two shorter subintervals at the end of the interval were allowed. Then Yu [21] gave new quadrature formulae to compute the case of singular point coinciding with the mesh point which presented that the error estimate is . In 1999, Wu and Yu [22] presented simple, easy to be implemented methods not affected by the location of singular point with calculation of double. In recent years, the case of singular point coincided with the mesh point, and Wu et al. [23] presented a modified trapezoidal rule and proved the convergence rate.

In this paper, for the case of singular point coinciding with the mesh point a new quadrature rule is introduced. Based on the expansion of the error functional, the error estimate is presented and a posteriori error estimate is given. Then not only do we obtain a high order of accuracy, but also a posteriori error estimate is conveniently derived.

The rest of this paper is organized as follows. In Section 2, after introducing some basic formulas of the general (composite) trapezoidal rule and notations, we present our main result. In Section 3, the corresponding theoretical analysis is given. Finally, several numerical examples are given to validate our analysis.

2. Main Result

Let be a uniform partition of the interval with mesh size and set

then we get the new partition:

We define , the modified trapezoidal interpolation for , as

and a linear transformation

from the reference element to the subinterval . For the two subintervals and near the end of the interval , values in and , respectively.

Replacing in (1) with gives the new composite trapezoidal rule:

where is the Cotes coefficients:

, denotes the Kronecker delta, and denotes the error functional.

Theorem 1. Assume , . For the trapezoidal rule defined in (7), there exists a positive constant , independent of and , such that

where

Proof. Let ; then we have , as

For the first part of (11), since , by Taylor expansion, we have

By the definition of finite-part integral,

we have

Now, we estimate the right hand side of (14) term by term. Since , we have

For the second part of (11),

Combining (14), (15), and (16) leads to (9) and the proof is completed.

Firstly, we set

then we have

By straight calculation, we get

Let and the operator be defined as

Obviously, the operator is linear operator. Then we set

Now we present our main results below. The proof will be given in the next section.

Theorem 2. Assume . For the trapezoidal rule defined in (7), there exists a positive constant , independent of and , such that

where , , and

( defined as (10)) and

3. Proof of Main Results

3.1. Preliminaries

In the following section, denotes certain constant independent of and , and its value varies with places.

Lemma 3. Assume that and are defined by (5); there holds

where

,,, and

Proof. Taking Taylor expansion for , at , there holds

by applying Taylor expansion to in (28) at , we have

Therefore, (25) can be obtained directly from (28) and (29).

Lemma 4. Assume that and ; there holds

Proof. According to (2) and the linear transformation (6) for , we have

where we have used . The second identity can be similarly obtained.

Lemma 5. Suppose and are defined by (18) and (24), respectively; then one has

where denotes that the first interval is certain part of the reference element.

Proof. By straightforward calculation, we have

Noting that , we have and

On the other hand, since , we have and

Combining (34) and (35), we get (32).

Set

Lemma 6. Under the same assumptions of Theorem 2, for in (36), there holds that

where is defined in (10).

Proof. Since , by Taylor expansion, we have

By the definition of finite-part integral (13), we have

Now, we estimate the right hand side of (39) term by term. Since , we have

Combining (40), (41), and (42) leads to (37) and the proof is completed.

Proof of Theorem 2. Consider

By the definition of , we have

Putting (43) and (44) together, we have

where

For the first part of ,

where and . We have also used the identity = , , and .

For the second part of ,

By Lemmas 5 and 6, we have

Then the proof is completed.

3.2. The Calculation of

Let be the function of the second kind associated with the Legendre polynomial , defined by (cf. [24])

We also define

Then, by the definition of ,

It follows that

which means

What remains is to determine the constant . By using the identities (cf. [24, Chapter 1, Section ]),

where denote the Bernoulli numbers, we have

Setting gives

Then we have

where we have used the formulae (cf. [24, Chapter 1, Section ])

Then we have

By Theorem 2, we get the following error expansion:

Let

where is the starting meshes, is the refining numbers, and . Then we have the following.

Corollary 7. Under the same assumption of Theorem 2 and (62), there holds

4. Numerical Examples

In this section, computational results are reported to confirm our theoretical analysis.

Example 1. Consider the hypersingular integral

We examine the dynamic point , in Table 1 show that when the local coordinate of singular point , the quadrature reach the convergence rate of as for the nonsupersingular point, there are no convergence rate which agree with our theorematically analysis. For the case of , Table 2 shows that there is no superconvergence phenomenon because of the influence of which coincides with our theoretical analysis.

Table 1: Errors of the mod-trapezoidal rule with .

32

64

128

256

512

1024

1.0239
1.9587
2.0111
1.0204

Table 2: Errors of the mod-trapezoidal rule with .

32

64

128

256

512

1024

1.0323
1.0054
1.0025
0.9857

Example 2. Consider the hypersingular integral

The numerical results show that the convergence rate reaches when the singular point coincides with the mesh point in Table 3. In Table 4, a posteriori error estimate is presented and the convergence rate is also which agrees with our theoretical analysis.

Table 3: Errors of the mod-trapezoidal rule.

32

64

1.9925

1.9844

1.9536

128

1.9963

1.9924

1.9781

256

1.9982

1.9963

1.9894

512

1.9991

1.9982

1.9948

1024

1.9996

1.9991

1.9974

Table 4: A posteriori error estimate of the mod-trapezoidal rule.

32

64

128

1.9912

1.9817

1.9452

256

1.9957

1.9911

1.9743

512

1.9979

1.9956

1.9875

1024

1.9989

1.9978

1.9939

Example 3. Now we consider an example of less regularity. Let , , and

Obviously, . The exact value of the integral is

The numerical results are presented in Tables 5 and 6. When the density function is smooth enough , the error bound is , and if the density function has less regularity , there is no hyperconvergence phenomenon, which means the regularity of density function cannot be reduced.

Table 5: Errors of the mod-trapezoidal rule.

32

1.1169

64

1.9481

1.4179

128

1.9659

1.4451

256

1.9770

1.4626

512

1.9843

1.4742

1024

1.9892

1.4820

Table 6: A posteriori error estimate of the mod-trapezoidal rule.

32

64

128

1.9420

1.4019

256

1.9621

1.4351

512

1.9746

1.4560

1024

1.9827

1.4698

Acknowledgments

The work of Jin Li was supported by the National Natural Science Foundation of China (nos. 11101247, 11201209, and 11101246), China Postdoctoral Science Foundation Fund Project (no. 2013M540541), the Shandong Provincial Natural Science Foundation of China (no. ZR2011AQ020), and a Project of Shandong Province Higher Educational Science and Technology Program (no. J11LE8).

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