A direct solver with O(N) complexity for integral equations on one-dimensional domains

Abstract

An algorithm for the direct inversion of the linear systems arising from Nyström discretization of integral equations on one-dimensional domains is described. The method typically has O(N) complexity when applied to boundary integral equations (BIEs) in the plane with non-oscillatory kernels such as those associated with the Laplace and Stokes’ equations. The scaling coefficient suppressed by the “big-O” notation depends logarithmically on the requested accuracy. The method can also be applied to BIEs with oscillatory kernels such as those associated with the Helmholtz and time-harmonic Maxwell equations; it is efficient at long and intermediate wave-lengths, but will eventually become prohibitively slow as the wave-length decreases. To achieve linear complexity, rank deficiencies in the off-diagonal blocks of the coefficient matrix are exploited. The technique is conceptually related to the H – and H ²-matrix arithmetic of Hackbusch and co-workers, and is closely related to previous work on Hierarchically Semi-Separable matrices.

Key words： Direct solver; integral equation; fast direct solver; boundary value problem; boundary integral equation; hierarchically semi-separable matrix

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