Here, the numerical solutions for Laplace equation with nonlinear boundary conditions is studied. Based on the potential theory, the problem can be converted into a nonlinear boundary integral equation. The modified quadrature method is presented for solving the nonlinear equation, which possesses high accuracy order $O(h^3)$ and low computing complexities. A nonlinear system is obtained by discretizing the nonlinear equation and the convergence of numerical solutions is proved by the theory of compact operators. Moreover, in order to solve the nonlinear system, the Newton iteration is provided by using the Ostrowski fixed point theorem. Finally, numerical examples support the theoretical results.

中文翻译：

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具有非线性边界条件的Laplace方程的改进正交方法

在此，研究了具有非线性边界条件的拉普拉斯方程的数值解。根据势能理论，可以将问题转换为非线性边界积分方程。提出了一种求解非线性方程的改进的正交方法，该方法具有较高的阶精度$ O（h ^ 3）$，计算复杂度低。通过离散非线性方程获得非线性系统，并通过紧算子理论证明了数值解的收敛性。此外，为了求解非线性系统，使用Ostrowski不动点定理提供了牛顿迭代。最后，数值算例支持了理论结果。