## Fast solution of boundary integral equations with the generalized Neumann kernel

Mohamed M.S. Nasser

### Abstract

A fast method for solving boundary integral equations with the generalized Neumann kernel and the adjoint generalized Neumann kernel is presented. The complexity of the developed method is $O((m+1)n\ln n)$ for the integral equation with the generalized Neumann kernel and $O((m+1)n)$ for the integral equation with the adjoint generalized Neumann kernel, where $m+1$ is the multiplicity of the multiply connected domain and $n$ is the number of nodes in the discretization of each boundary component. Numerical results illustrate that the method gives accurate results even for domains of very high connectivity, domains with piecewise smooth boundaries, domains with close-to-touching boundaries, and domains of real world problems.

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### Key words

generalized Neumann kernel, boundary integral equations, Nyström method, Fast Multipole Method, GMRES, numerical conformal mapping

### AMS subject classifications

45B05, 65R20, 30C30

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