A Fast Fourier-Galerkin Method for Solving Singular Boundary Integral Equations

Authors:
Haotao Cai;Yuesheng Xu
Affiliations:
caihaotao@126.com;yxu06@syr.edu
Venue:
SIAM Journal on Numerical Analysis
Year:
2008

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Abstract

We propose in this paper a convenient way to compress the dense matrix representation of a compact integral operator with a smooth kernel under the Fourier basis. The compression leads to a sparse matrix with only ${\cal O}(n\log n)$ nonzero entries, where $2n$ or $2n+1$ denotes the order of the matrix. Based on this compression strategy, we develop a fast Fourier-Galerkin method for solving a class of singular boundary integral equations. We prove that the fast Fourier-Galerkin method gives the optimal convergence order ${\cal O}(n^{-t})$, where $t$ denotes the degree of regularity of the exact solution. Moreover, we design a fast scheme for solving the corresponding truncated linear system. We show that solving this system requires only an ${\cal O}(n\log^2 n)$ number of multiplications. We present numerical examples to confirm the theoretical estimates and to demonstrate the efficiency and accuracy of the proposed algorithm.