Abstract
The composite trapezoidal rule for the computation of Cauchy principal value integral with the singular kernel is discussed. Our study is based on the investigation of the pointwise superconvergence phenomenon; that is, when the singular point coincides with some a priori known point, the convergence rate of the trapezoidal rule is higher than what is globally possible. We show that the superconvergence rate of the composite trapezoidal rule occurs at middle of each subinterval and obtain the corresponding superconvergence error estimate. Some numerical examples are provided to validate the theoretical analysis.
1. Introduction
Consider the Cauchy principal integral where denotes a Cauchy principal value integral and is the singular point.
There are several different definitions which can be proved equally, such as the definition of subtraction of the singularity, regularity definition, and direct definition.
In this paper we adopt the following one:Cauchy principal value integral has recently attracted a lot of attention; see, for example, [1]. The main reason for this interest is probably due to the fact that integral equations with Cauchy principal value integrals have been shown to be an adequate tool [1] for the modeling of many physical situations, such as acoustics, fluid mechanics, elasticity, fracture mechanics, and electromagnetic scattering problems. Numerous work has been devoted in developing efficient quadrature formulas, such as the Gaussian method [2–8], the Newton-Cotes methods [9–13], spline methods [14, 15], and some other methods [16–22].
It is the aim of this paper to investigate the superconvergence phenomenon of trapezoidal rule and, in particular, to derive error estimates. In this paper the density function is replaced by the approximation function while the singular kernel is computed in each subinterval, where is the trapezoidal rule. This method is different from the semidiscrete methods and the order of singularity kernel can be reduced somehow which was firstly presented by Linz in the paper to calculate the hypersingular integral on interval. He used the trapezoidal rule and Simpson rule to approximate the density function and the convergence rate was , , when the singular point is always located at the middle of certain subinterval.
The superconvergence of composite Newton-Cotes rules for Hadamard finite-part integrals was studied in [23, 24], where the superconvergence rate and the superconvergence point were presented, respectively. In [25, 26] the classical composite midpoint rectangle rule and classical composite trapezoidal rectangle rule for the computation of Cauchy principal value integrals are discussed. When the singular point coincides with some a priori known point, the convergence rate of the midpoint rectangle rule is higher than the global one, the same as the Riemann integral, which is called superconvergence phenomenon.
This paper focuses on the superconvergence of trapezoidal rule for Cauchy principle integrals in a circle. Based on the investigation of the pointwise superconvergence phenomenon, that is, when the singular point coincides with some a priori known point, the convergence rate of the trapezoidal rule is higher than what is globally possible. We show that the superconvergence rate of the composite trapezoidal rule occurs at middle of each subinterval and obtain the corresponding superconvergence error estimate. We prove both theoretically and numerically that the composite trapezoidal rule reaches the superconvergence rate when the local coordinate of the singular point is 0.
The rest of this paper is organized as follows. In Section 2, after introducing some basic formulas of the rectangle rule, we present the main results. In Section 3 we perform the proof. Finally, several numerical examples are provided to validate our analysis.
2. Main Result
Let be a uniform partition of the interval with mesh size . Define by the piecewise line interpolant for :and a linear transformationfrom the reference element to the subinterval . Replacing in (2) with gives the composite trapezoidal rule:where denote the Cotes coefficients given by
We also define
Theorem 1. Assume . For the trapezoidal rule is defined as (5). Assuming that , there exists a positive constant , independent of and , such thatwhere
Proof. Let ; then we have . ThusFor the first part of (10), we haveFor the second part of (10), we haveFor the third part of (10), we haveCombining (11), (13), and (14) together, the proof is completed.
Set
Lemma 2. Assume with . Let be defined by (15); then there holds that
Proof. For , by the definition of Cauchy principal value integral, we haveFor , taking integration by parts on the correspondent Riemann integral, we haveNow, by using the well-known identity,The proof is completed.
Lemma 3. Under the same assumptions of Lemma 2, there holds that
Proof. By (15), we haveThe proof is completed.
Theorem 4. Assume . For the trapezoidal rule is defined as (5). Assuming that , there exists a positive constant , independent of and , such thatwhere is defined as (9).
It is known that the global convergence rate of the composite trapezoidal rule is lower than Riemann integral.
3. Proof of the Theorem
In this section, we study the superconvergence of the composite trapezoidal rule for Cauchy principle integrals.
3.1. Preliminaries
In the following analysis, will denote a generic constant that is independent of and and it may have different values in different places.
Lemma 5. Under the same assumptions of Theorem 4, it holds that where
Proof. Performing Taylor expansion of at the point , we haveSimilarly, we haveWe set
Lemma 6. Let , denote to be the error functional for the composite trapezoidal rule, and assume for any ; then there holds
Proof. As , we get . We haveFor the first part of (30), by the definition ofthen we havethen we getFor the second part of (30), we haveAnd the proof is completed.
Proof of Theorem 4. By Lemma 5, we haveBy the definition of , we havePutting (35), (36) together yieldswhereNow we estimate term by term. For the first part of , we haveFor the second part of , there is no singularity and we haveThe third and fourth parts of can be similarly obtained. For , by Lemmas 5 and 6, then we getand the proof is completed.
Remark 7. By the identity in [27],then we getWe can prove that the error expansion of this paper has the same error expansion of [25] for the Cauchy principal integral defined on the interval. Based on the error expansion, we get the same superconvergence point with local coordinate point equal to zero.
Based on Theorem 4, we present the modified trapezoidal rule
4. Numerical Example
In this section, computational results are reported to confirm our theoretical analysis.
Example. We consider the singular integral with . and with are the superconvergence point.
From Tables 1 and 2, we know that the superconvergence point is 0 with the coordinate location of singular point equal to zero, while, for the local coordinate of singular point not equal to zero, it is not convergence in general which coincides with our analysis.
For the modified classical trapezoidal rule, from Tables 3 and 4, for the nonsuperconvergence point and the superconvergence point, we all get the superconvergence phenomenon.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The work of Jin Li was supported by the National Natural Science Foundation of China (Grant no. 11471195, Grant no. 11101247, and Grant no. 11201209) and China Postdoctoral Science Foundation (Grant nos. 2013M540541 and 2015T80703).