Uniqueness and convergence of numerical solution of the Cauchy problem for the Laplace equation by a charge simulation method

We consider the Cauchy problem for the Laplace equation in the neighborhood of the circle. The charge simulation method is applied to the problem, and a theoretical analysis for the numerical solution is given. The analysis for the numerical solution of the charge simulation method can be found in some papers, but the approach in these papers is only for well-posed problems such as the Dirichlet problem. Since our problem is ill-posed, a different approach is required to analyze the convergence of the numerical solution. In this paper, we prove the unique existence of the numerical solution and its exponential convergence to the exact solution. Our result agrees well with numerical experiments.