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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 singu- lar integrals in the equation. The first method uses the product Simpson's rule, and the second is based on trigonometric interpolation. Iterative methods (in- cluding two-grid methods) for solving the resulting nonlinear systems are also discussed extensively. Numerical examples are included.

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