The numerical solution of first-kind logarithmic-kernel integral equations on smooth open arcs

Atkinson, Kendall E.; Sloan, Ian H.

doi:10.1090/S0025-5718-1991-1052084-0

The numerical solution of first-kind logarithmic-kernel integral equations on smooth open arcs
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by Kendall E. Atkinson and Ian H. Sloan PDF

Math. Comp. 56 (1991), 119-139 Request permission

Abstract:

Consider solving the Dirichlet problem \[ \begin {array}{*{20}{c}} {\Delta u(P) = 0,} \hfill & {P \in {\mathbb {R}^2}\backslash S,} \hfill \\ {u(P) = h(P),} \hfill & {P \in S,} \hfill \\ {\sup |u(P)| < \infty ,} \hfill & {} \hfill \\ {P \in {\mathbb {R}^2}} \hfill & {} \hfill \\ \end {array} \] with S a smooth open curve in the plane. We use single-layer potentials to construct a solution $u(P)$. This leads to the solution of equations of the form \[ \int _S {g(Q)\log |P - Q|dS(Q) = h(P),\quad P \in S.} \] This equation is reformulated using a special change of variable, leading to a new first-kind equation with a smooth solution function. This new equation is split into a principal part, which is explicitly invertible, and a compact perturbation. Then a discrete Galerkin method that takes special advantage of the splitting of the integral equation is used to solve the equation numerically. A complete convergence analysis is given; numerical examples conclude the paper.

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