A meshless discrete Galerkin (MDG) method for the numerical solution of integral equations with logarithmic kernels

This paper describes a computational method for solving Fredholm integral equations of the second kind with logarithmic kernels. The method is based on the discrete Galerkin method with the shape functions of the moving least squares (MLS) approximation constructed on scattered points as basis. The MLS methodology is an effective technique for the approximation of an unknown function that involves a locally weighted least square polynomial fitting. The numerical scheme developed in the current paper utilizes the non-uniform Gauss-Legendre quadrature rule for approximating logarithm-like singular integrals and so reduces the solution of the logarithmic integral equation to the solution of a linear system of algebraic equations. The proposed method is meshless, since it does not require any background mesh or domain elements. The error analysis of the method is provided. The scheme is also applied to a boundary integral equation which is a reformulation of a boundary value problem of Laplace's equation with linear Robin boundary conditions. Finally, numerical examples are included to show the validity and efficiency of the new technique.