RESEARCH ARTICLE

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

  • Adrianna GILLMAN ,
  • Patrick M. YOUNG ,
  • Per-Gunnar MARTINSSON
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  • Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO 80309-0526, USA

Received date: 21 Feb 2011

Accepted date: 28 Aug 2011

Published date: 01 Apr 2012

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

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.

Cite this article

Adrianna GILLMAN , Patrick M. YOUNG , Per-Gunnar MARTINSSON . A direct solver with O(N) complexity for integral equations on one-dimensional domains[J]. Frontiers of Mathematics in China, 2012 , 7(2) : 217 -247 . DOI: 10.1007/s11464-012-0188-3

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