Fast multipole method for singular integral equations of second kind
- Qinghua Wu^{1}Email author and
- Shuhuang Xiang^{2}
https://doi.org/10.1186/s13662-015-0515-6
© Wu and Xiang 2015
Received: 5 December 2014
Accepted: 21 May 2015
Published: 20 June 2015
Abstract
This paper explores the numerical methods for a singular integral equation (SIE), which arise in the study of various problems of mathematical physics and engineering. The idea behind the boundary element method (BEM) is used to discretize the SIE. The fast multipole method (FMM), which is a very efficient and popular algorithm for the rapid solution of boundary value problems, is used to accelerate the BEM solutions of SIE. The effectiveness and accuracy of the proposed method are tested by numerical examples.
Keywords
1 Introduction
In general, it will give rise to a standard linear system of equations when approximating the solution of SIE by numerical methods. Specially, collocation methods would lead to systems of equations with dense and non-symmetrical coefficient matrices where \(O(N^{2})\) elements need to be stored, with N being the number of degrees of freedom. Solving the systems of equations directly will need \(O(N^{3})\) arithmetic operations. Fortunately, Rokhlin and Greengard innovated the fast multipole method (FMM) which has been widely used for solving large scale engineering problems such as potential, elastostatic, Stokes flow, and acoustic wave problems. For one dimension (SIE (1)), the interval \([-1,1]\) is not a closed curve; however, we can also utilize BEM to solve SIE and accelerate BEM by FMM when the kernel \(k(y,x)\) has multipole expansion or \(k(y,x)= 0\), for details, see [18–20].
We approximate the solution of SIE by the collocation methods and utilize the FMM to improve the efficiency of algorithm. The paper is organized as follows. In Section 2 we give a brief description of the FMM, where the multipole expansion theory is introduced and also moment to moment translation (M2M), moment to local translation (M2L), and local to local translation (L2L). In Section 3, we give the convergence analysis of the proposed method. In Section 4 we give preliminary numerical examples to illustrate the effectiveness and accuracy of the proposed method.
2 Fast multipole boundary element method for the solution of (2)
2.1 Multipole expansion of the kernel
To derive the multipole expansion of kernel \(G(y,x):=\frac {1}{(y-x)^{m}}\), we need the following formulation.
Lemma 1
([22])
2.2 Evaluation of the integrals
3 Convergence analysis
In this section, we derive the error bound for (10) when \(m=1\).
Lemma 2
([23])
Lemma 3
([24])
Theorem 1
Proof
Theorem 2
Proof
The desired error bound is established by setting \(M= \frac{1}{h}+(p+1)M_{1}C_{1}\). □
Remark 1
In the FMM, \(x_{l}\) and \(y_{c}\) are well-separated points, we could obtain \(\vert y_{c}-x_{l}\vert > 2h\), which leads to \(C_{1}\) is bounded.
4 Numerical examples
We illustrate the efficiency and accuracy of the methods described in this paper by numerical examples. Here \(\hat{u}_{N}\) denotes the piecewise constant collocation method, N is the number of collocation points. We choose the uniform mesh, and \(\tilde{x}_{i}\) is the middle point of \((x_{i-1},x_{i})\). Denote by \(I_{h}^{N}=\{\tilde{x}_{i},i=1,\ldots,N\}\) the set of collocation points.
Example 4.1
Approximations at \(\pmb{x=-0.9,-0.5,-0.1}\) for \(\pmb{u(x)-\frac{1}{2\pi}\int_{-1}^{1}\frac{u(y)}{(y-x)}\,dy=f(x)}\)
x | −0.9 | −0.5 | −0.1 |
---|---|---|---|
\(\hat{u}_{10}\) | 1.000000000008664 | 0.999999999966656 | 1.000000002837690 |
\(\hat{u}_{100}\) | 0.999999988989405 | 0.999999996983473 | 0.999999998141215 |
\(\hat{u}_{1000}\) | 1.000000006379357 | 0.999999997488601 | 0.999999998811889 |
u | 1 | 1 | 1 |
Approximations at \(\pmb{x=0.1,0.5,0.9}\) for \(\pmb{u(x)-\frac{1}{2\pi}\int_{-1}^{1}\frac{u(y)}{(y-x)}\,dy=f(x)}\)
x | 0.1 | 0.5 | 0.9 |
---|---|---|---|
\(\hat{u}_{10}\) | 0.999999996078344 | 1.000000000199640 | 1.000000000023095 |
\(\hat{u}_{100}\) | 1.000000000689915 | 1.000000003352249 | 1.000000033188974 |
\(\hat{u}_{1000}\) | 1.000000000150622 | 1.000000002673433 | 0.999999995294152 |
u | 1 | 1 | 1 |
Approximations at \(\pmb{x=-0.9,-0.5,-0.1}\) for \(\pmb{u(x)-\frac{1}{2\pi}\int_{-1}^{1}\frac{u(y)}{(y-x)^{2}}\,dy=f(x)}\)
x | −0.9 | −0.5 | −0.1 |
---|---|---|---|
\(\hat{u}_{10}\) | 0.999999993535333 | 0.999999981428145 | 0.999999917999549 |
\(\hat{u}_{100}\) | 0.999999993845970 | 0.999999988715871 | 0.999999990115998 |
\(\hat{u}_{1000}\) | 0.999999995690078 | 0.999999991935489 | 0.999999993153335 |
u | 1 | 1 | 1 |
Approximations at \(\pmb{x=0.1,0.5,0.9}\) for \(\pmb{u(x)-\frac{1}{2\pi}\int_{-1}^{1}\frac{u(y)}{(y-x)^{2}}\,dy=f(x)}\)
x | 0.1 | 0.5 | 0.9 |
---|---|---|---|
\(\hat{u}_{10}\) | 0.999999917999549 | 0.999999981428145 | 0.999999993535332 |
\(\hat{u}_{100}\) | 0.999999990036579 | 0.999999988582430 | 0.999999992194477 |
\(\hat{u}_{1000}\) | 0.999999993375294 | 0.999999992061794 | 0.999999995726361 |
u | 1 | 1 | 1 |
From Tables 1-4, it is easy to see that the proposed method is effective. It might also be noted that \(\tilde{x}_{i} \in I_{h}^{10}\) but \(\tilde {x}_{i} \notin I_{h}^{100}\) and \(\tilde{x}_{i} \notin I_{h}^{1000}\), i.e., the points \(\{-0.9,-0.5,-0.1,0.1, 0.5,0.9 \}\subset I_{h}^{10}\), but it is not a subset of \(I_{h}^{100}\) or \(I_{h}^{1000}\).
Example 4.2
Approximations at \(\pmb{x=-0.9,-0.5,-0.1}\) for \(\pmb{u(x)-\int_{-1}^{1}\frac{u(y)}{(y-x)}\,dy=f(x)}\)
x | −0.9 | −0.5 | −0.1 |
---|---|---|---|
\(\hat{u}_{10}\) | −1.194519628297830 | −0.601189820467772 | −0.156767445549832 |
\(\hat{u}_{100}\) | −0.902665658196492 | −0.496591018161909 | −0.094652542268498 |
\(\hat{u}_{1000}\) | −0.900369341742219 | −0.499656299326138 | −0.099458627117756 |
u | −0.9 | −0.5 | −0.1 |
Approximations at \(\pmb{x=0.1,0.5,0.9}\) for \(\pmb{u(x)-\int_{-1}^{1}\frac{u(y)}{(y-x)}\,dy=f(x)}\)
x | 0.1 | 0.5 | 0.9 |
---|---|---|---|
\(\hat{u}_{10}\) | 0.057786709933708 | 0.465859007671879 | 0.872182894009698 |
\(\hat{u}_{100}\) | 0.106011730395414 | 0.507142753891225 | 0.908324232818361 |
\(\hat{u}_{1000}\) | 0.100609029282889 | 0.500725569118200 | 0.900861403735125 |
u | 0.1 | 0.5 | 0.9 |
Example 4.3
Approximations at \(\pmb{x=-0.9,-0.5,-0.1}\) for \(\pmb{u(x)-\int_{-1}^{1}\frac{u(y)}{(y-x)}\,dy=f(x)}\)
x | −0.9 | −0.5 | −0.1 |
---|---|---|---|
\(\hat{u}_{10}\) | −0.608896542845208 | −0.143953205178288 | −0.019345651255813 |
\(\hat{u}_{100}\) | −0.400907820963934 | −0.096826636160551 | −0.002238964950304 |
\(\hat{u}_{1000}\) | −0.402641823825626 | −0.099675717661578 | −0.001102497056280 |
u | −0.402762430939227 | −0.100000000000000 | −0.000990099009901 |
Approximations at \(\pmb{x=0.1,0.5,0.9}\) for \(\pmb{u(x)-\int_{-1}^{1}\frac{u(y)}{(y-x)}\,dy=f(x)}\)
x | 0.1 | 0.5 | 0.9 |
---|---|---|---|
\(\hat{u}_{10}\) | −0.025111748216482 | 0.052110055810136 | 0.363240467300408 |
\(\hat{u}_{100}\) | −0.001202605523378 | 0.101502077326766 | 0.409787561038363 |
\(\hat{u}_{1000}\) | 0.000783503911385 | 0.100161733460852 | 0.403492522553747 |
u | 0.000990099009901 | 0.1 | 0.402762430939227 |
5 Conclusion
In this paper, we explore collocation methods for a singular Fredholm integral equation of the second kind and utilize the FMM to improve the efficiency of algorithm. Based on the multipole expansion of kernel, we demonstrate that the approximate operator used in the collocation equation converges to the initial operator. Numerical examples demonstrate the performance of the proposed algorithm.
Declarations
Acknowledgements
The authors are grateful to the anonymous referees for their constructive comments and helpful suggestions to improve this paper greatly. This work is supported by the Scientific Research Foundation of Education Bureau of Hunan Province under grant 14C0495, the Natural Science Foundation of Hunan Province of China under grant 14JJ3134, the NSF of China under grant 11371376, the Mathematics and Interdisciplinary Sciences Project of Central South University.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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