Extrapolation Algorithm for Computing Multiple Cauchy Principal Value Integrals

In this paper, the computation of multiple (including two dimensional and three dimensional) Cauchy principal integral with generalized composite rectangle rule was discussed, with the density function approximated by themiddle rectangle rule, while the singular kernel was analytically calculated. Based on the expansion of the density function, the asymptotic expansion formulae of error functional are obtained. A series is constructed to approach the singular point, then the extrapolation algorithm is presented, and the convergence rate is proved. At last, some numerical examples are presented to validate the theoretical analysis.


Introduction
With the development of boundary element methods, a lot of attentions had recently been paid to Cauchy principal value integrals (including one dimensional and two dimensional). In this paper, we pay our attention to certain two-dimensional Cauchy principal value integral of the form where L 1 L 2 denotes a Cauchy Principle value integral L 1 � [a, b], L 2 � [c, d], and (t, s) ∈ (a, b) × (c, d) the singular point.
In [1], based on the Gauss quadrature of one-dimensional Cauchy principal value integral with weight function ω(x) � (1 − x) α (l + x) β , α, β > − 1, generalized quadrature Gauss rule for two-dimensional Cauchy principal value integrals is presented. In [23], for the problem of computing 2D Cauchy principal value integrals of the form S F(P 0 , P)dP, authors have constructed quadrature rules of the Gaussian type. In [3], for the two-dimensional Cauchy principal value integrals with respect to generalized smooth Jacobi weight functions, authors have considered product rules of the Gauss type for the numerical approximation of two-dimensional Cauchy principal value integrals. In [24], a principal value definition of the basic hypersingular integral in the fundamental integral equation for two-dimensional cracks in three-dimensional isotropic elasticity is proposed. In [25], the classical composite rectangle (midpoint) rule for the computation of two-dimensional singular integrals is discussed, with the error functional of the rectangle rule for computing two-dimensional singular integrals, and the local coordinate of certain point and the convergence results O(h 2 ) are obtained. In [26], the classical composite trapezoidal rule for the computation of two-dimensional singular integrals is presented and the convergence results O(h 2 ) is the same as the Riemann integral convergence rate at a certain point of the classical composite trapezoidal rule. In [27], the Cauchy principal value integral is approximated by the zeros of the Chebyshev polynomials of the first kind. In [28], a twelve-point cubature formula has been constructed for the numerical evaluation of two-dimensional real Cauchy principal value integrals. In [29], Gaussian quadrature rules for the evaluation of Cauchy principal values of integrals are considered and their relation with Gauss-Legendre formulae is also studied. In [30], the density function f(x, y) was presented as en, (1) can be written as which is the method with subtraction of singularity while the singular part was calculated analytically. In [31,32], Lyness studied the Euler-Maclaurin expansion technique for the evaluation of two-dimensional Cauchy principal integrals. e extrapolation methods based on Hadamard finite-part integral definition with the trapezoidal rule for the computation of hypersingular integrals on interval and in circle in boundary element methods are presented in [33,34], respectively. e extrapolation methods based on definition of subtraction of singularity are presented in [35,36], respectively.
In this paper, we focus on certain kinds of two-dimensional Cauchy principal value integrals which have not been extensively studied. It is the aim of this paper to investigate the error expansion of the rectangle rule and, in particular, to derive the error estimates. We examine the convergence property of the rectangle rule for certain kinds of the two-dimensional Cauchy principal value integrals and generalize the abovementioned one-dimensional convergence results to cover this new situation. Moreover, we give an error expansion of the corresponding remainder when the density function f(x, y) belongs to As the special function of the error expansion equals zero, we get the superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate is higher than what is globally possible. en, the extrapolation algorithm is presented and the convergence rate of an extrapolation error and a posterior error is proved. e rest of this paper is organized as follows. In Section 2, after introducing some basic formulae of the rectangle rule, we present the main results. In Section 3, we finished the proof. Finally, several numerical examples are provided to validate our analysis.

Main Result
In order to simplify our analysis in the following, we set h � h x � h y � (b − a)/n � (d − c)/n; it is not difficult to extend our analysis to the case of quasi-uniform meshes.
We set x i � x i + h/2, y j � y j + h/2, i, j � 0, 1, . . . , n − 1, and define f C (x, y) as the piecewise constant interpolation for f(x, y): and then we obtain where E n (f, t, s) denotes the error functional and the Cotes coefficient is Define the following linear transformation: which can be expressed as the Legendre function of second kinds in [37]. We also set J ≔ (x, y) ∈ R 2 : |x| ≠1 and |y| ≠ 1 , and the operator W: C(J) ⟶ C(− 1, 1)× (− 1, 1) can be defined by en, we have e superconvergence results of constant rectangle rules are given in the following. [20]). Let S 0x (τ, ξ) and S 0y (τ, ξ) be defined by (12) and (13), respectively. Assume that f(x, y) ∈ C 2 [a, b] × [c, d] and t ≠ x i , s ≠ y j , for any i, j � 0, 1, . . . , n. For the middle rectangle rule (6), there exists a positive constant C, independent of h, t, and s, such that
In the following theorem, we present the asymptotic expansion formulae of the error functional. e proof of this theorem will be given in the next part.

Proof of Theorem 2
In this section, we study the asymptotic expansion of the composite rectangle rule for the multiple (two dimensional) Cauchy principle value integrals.

Preliminaries.
In the following analysis, C will denote a generic constant that is independent of h and s, and it may have different values in different places.
Mathematical Problems in Engineering Proof. In the case of i � k and j � l, we have en, we prove the first identity in (19). e second identity in (19) can be obtained by applying the approach to the correspondent Riemann integral. e two identities in (20) can be obtained similarly as (19).

□
Proof of eorem 2. By Taylor expansion, taking f C (x, y) and the point (x, y), we obtain For the case i � k and j � l, we have Putting (22) and (23) where we have used the Taylor expansion of f(x, y) at the point (t, s):

Mathematical Problems in Engineering
Here, we have Similarly, we can have As for the part of which can be considered as the error estimate of the rectangle rule for the definite integral It is easy to see that there is no relation with the singular point (t, s) and the Proof of eorem 2 is completed.
It is not difficult to extend our analysis to the case of multidimensional Cauchy principal value integrals. Let We set x ki � x ki + h/2, i � 0,1, ... , n − 1, and define f C (x k1 , x k2 , .. ., x kn ) as the piecewise constant interpolation for f(x, y): en, we obtain Mathematical Problems in Engineering 9 where E n (f; t 1 , t 2 , . . . , t k ) denotes the error functional and the Cotes coefficient is For the rectangle rule I n (f; t 1 , t 2 , . . . , t k ) defined in (36), there exist certain constant a i 1 ,...,i k (τ 1 , . . . , τ k ), independent of h and t 1 , t 2 , . . . , t k , such that where l≥ 1,

Extrapolation Method
In the abovementioned sections, we have proved that the error functional of the middle rectangle rule has the following asymptotic expansion: It is easy to see that the error functional depends on the value of a i,j (τ, ξ). Now we present an algorithm for the given (t, s). Assume there exist positive integers n 01 and n 02 such that are positive number. In order to simplify the deduction process, we assume that n 10 � m 20 . Firstly, we partition [a, b] × [c, d] into n 01 equal subinterval to get a mesh denoted by Π 1 with mesh size h 1 � (b − a)/n 10 � (d − c)/n 20 . en, we refine Π 1 to get mesh Π 2 with mesh size h 2 � h 1 /2. In this way, we get a series of meshes Π j (j � 1, 2, . . .) in which Π j is refined from Π j− 1 with the mesh size denoted by h j . e extrapolation scheme is presented in Table 1.

Theorem 4. Under the asymptotic expansion of eorem 2, for a given τ, ξ and the series of meshes defined by (42), we have
and a posteriori asymptotic error estimate is given by Proof. By the asymptotic expansion of (40), for given τ, we have By the definition of Cauchy principal integral and (42), for the first two parts of (48) by Taylor expansion for I(f, t i , s j ) at the point (t, s), j�0,1,...,l (49) Similarly we also expand f (i,j) (t i , s j ) at point (t, s), and then we have Putting (48)-(50) together, we have s, τ, ξ) is a constant for a given τ and ξ. By (51), we can obtain By equations (51) and (53), with h j � 2h j+1 , we easily have which means Following the extrapolation process above, we can get the accuracy O(h 3 ), and we continue to use extrapolation process again and get O(h 4 ). In this way, we finish the proof by mathematical induction.

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

Example 1. Consider the Cauchy principal value integral
with the exact solution We adopt the uniform meshes to examine the convergence rate of the rectangle rule with the dynamic point with t � x [n/4] + (1 + τ)h/2 and s � y [n/4] + (ξ + 1)h/2. From Table 2, we know that when the local coordinate of the singular point is ( ± (2/3), ± (2/3)), the quadrature reaches the convergence rate of O(h 2 ); as for the nonsupersingular point, the convergence rate is O(h) which agree with our theoretical analysis. For the case of t � x [n/4] + (τ + 1)h/2 and s � b − (τ + 1)h/2 and t � a + (τ + 1)h/2 ands � b − (τ + 1)h/2, Tables 3 and 4 show that there is no superconvergence phenomenon for the midrectangle rue which coincide with our theoretical analysis.
Example 2. In this example, we still consider the Cauchy principal value integral with the exact solution with the exact solution Mathematical Problems in Engineering               O(h 3 ), . . . , with τ 1 � τ 2 � τ 3 � 2/3, which agrees with our eorem 4.

Conclusion
In this paper, we have shown, both theoretically and numerically, that the main part of error functional of the rectangle rule has the asymptotic expansion (18). Numerical experiment has shown that the special function a i,j (τ, ξ) has a big influence on the convergence rate. Moreover, by the extrapolation method, we not only obtain a high order of accuracy but also derive a posteriori error estimate conveniently. In fact, it is not difficult to extend our results to arbitrary (multidimensional) Cauchy principal value integral and the extrapolation methods can be similarly obtained.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.