Petrov-Galerkin Method on Modified Symm's Integral Equation from Interior and Exterior Dirichlet Problem

In two dimension, for bounded simply connected domain $ \Omega $ of infinitely smooth class, the boundary value problem of Laplace equation can be transformed into Symm's integral equation, the convergence and divergence analysis are done respectly. In this paper, we follow the procedure to handle the modified Symm's integral equation which solves the interior and exterior Dirichlet problem of Laplace equation, give the convergence and divergence result, including optimal convergence rate and divergence rate.