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.
Highlights
Various problems of mathematical physics and engineering can be described by differential equations which can often be reformulated as an equivalent integral equation
Xiang and Brunner [, ] introduced collocation and discontinuous Galerkin methods for Volterra integral equations with highly oscillatory Bessel kernels, and they concluded that the collocation methods are much more implemented and can get higher accuracy than discontinuous Galerkin methods under the same piecewise polynomials space
We approximate the solution of singular integral equation (SIE) by the collocation methods and utilize the fast multipole method (FMM) to improve the efficiency of algorithm
Summary
Various problems of mathematical physics and engineering can be described by differential equations which can often be reformulated as an equivalent integral equation. Many numerical methods have been developed to solve SIE ( ), among which collocation methods, Galerkin methods, spectral methods, etc. It will give rise to a standard linear system of equations when approximating the solution of SIE by numerical methods. 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. We approximate the solution of SIE by the collocation methods and utilize the FMM to improve the efficiency of algorithm. Due to matrix-vector multiplication, solving this system by iterative solvers such as the generalized minimum residue (GMRES) method needs O(N ) operations, and even worse by direct solvers such as Gauss elimination. Based on the multipole expansion of the kernel, the FMM can be used to accelerate the matrix-vector multiplication
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