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A Nyström-Like Approach to Integral Equations with Singular Kernels

S.-H. Kuo and J.K. White
Massachusetts Institute of Technology, US

Keywords:
integral equation, Nystrom method, spectral convergence

Abstract:
When boundary element methods are used to solve Laplace or Helmholtz problmes associated with complicated three-dimensional geometries, the associated integral equation is typically discretized using a piecewise constant basis, and a system of equations is generated using either a Galerkin or a collocation scheme. The resulting matrix is then solved iteratively using acceleration. This approach has become the method of choice for exterior problems and enjoys success in applications such as interconnect extraction, MEMS and fluidic simulation, as well as in calculating bimolecular solvation energy. However, piecewise-constant bases are low order, and therefore large numbers of unknowns are needed to achieve high accuracy. While acceleration techniques make it possible to solve such problems, memory is often a bottleneck. Therefore, there is much interest in developing higher order methods that can achieve faster convergence and reduce problem size. In this paper, we propose a new kind of higher order basis and demonstrate spectral convergence (error decays exponentially with number of unknowns).

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