Boundary integral equation methods for solving Laplace’s equation with nonlinear boundary conditions: the smooth boundary case

Atkinson, Kendall E.; Chandler, Graeme

doi:10.1090/S0025-5718-1990-1035924-X

Boundary integral equation methods for solving Laplace’s equation with nonlinear boundary conditions: the smooth boundary case
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by Kendall E. Atkinson and Graeme Chandler PDF

Math. Comp. 55 (1990), 451-472 Request permission

Abstract:

A nonlinear boundary value problem for Laplace’s equation is solved numerically by using a reformulation as a nonlinear boundary integral equation. Two numerical methods are proposed and analyzed for discretizing the integral equation, both using product integration to approximate the singular integrals in the equation. The first method uses the product Simpson’s rule, and the second is based on trigonometric interpolation. Iterative methods (including two-grid methods) for solving the resulting nonlinear systems are also discussed extensively. Numerical examples are included.

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Trigonometric series