Abstract
We propose a fast algorithm for the solution of the nonlinear boundary integral equation resulting from a reformulation of a boundary value problem of the Laplace equation with nonlinear boundary conditions. The fast algorithm is developed by using the multilevel augmentation method (introduced recently by Chen, Wu, and Xu for general nonlinear integral equations), in conjunction with a matrix truncation strategy, and an error control technique of numerical integrations for integrals appeared in the process of solving the equation. We prove that the proposed algorithm has an optimal convergence order (up to a logarithmic factor) and a nearly linear computational complexity order (measured in the number of multiplications and functional evaluations). Numerical experiments are presented to demonstrate its approximation accuracy and computational efficiency, verifying the theoretical estimates, and to compare performance of the proposed algorithm with that of the Atkinson and Chandler algorithm.
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