Analysis of the inherent instability of the interpolating moving least squares method when using improper polynomial bases

This paper first discusses the inherent instability of the interpolating moving least squares (IMLS) method. In the original IMLS method, non-scaled polynomial bases are used. Theoretical and numerical results indicate that the stability of the original IMLS method decreases as the separation distance decreases. Then, using shifted and scaled polynomial bases, a stabilized algorithm of the IMLS method is proposed and analyzed. As an application, the stabilized IMLS method is finally introduced into the meshless Galerkin boundary node method (GBNM) to produce a stabilized GBNM for potential problems and Stokes problems. Numerical examples are given to demonstrate the stability and convergence of the presented stabilized algorithms.