Analytical diagonal elements of regularized meshless method for regular domains of Dirichlet Laplace problems

This note is to present a simple approach to derive the analytical formula of the diagonal elements of the interpolation matrix of the regularized meshless method (RMM) for regular domain problems, which is a very new boundary-type numerical discretization technique. In literature, these diagonal elements are mostly calculated numerically by the desingular technique, except for the circular domain problems. Our numerical experiments show that the analytical diagonal elements can improve the solution accuracy of the RMM for some regular domain problems, and the diagonal elements are critical to the solution accuracy of the RMM. Thus, a searching process is employed to find the optimal diagonal elements for RMM.