A new spectral meshless radial point interpolation (SMRPI) method: A well-behaved alternative to the meshless weak forms

Abstract

In this article, a new spectral meshless radial point interpolation (SMRPI) technique is proposed and, as a test problem, applied to the two-dimensional diffusion equation with an integral (non-classical) condition. This innovative method is based on erudite combination of meshless methods and spectral collocation techniques but it is not traditional meshless collocation method. As the meshless method, the point interpolation method with the help of radial basis functions is used to construct shape functions as basis functions in the frame of spectral collocation methods. These basis functions (shape functions) have Kronecker delta function property. In the proposed method, high order derivatives (in high order partial differential equations) are evaluated by constructing and using operational matrices. In SMRPI method, it does not require any kind of integration locally over small quadrature domains as it is essential in Galerkin weak form meshless methods such as element free Galerkin (EFG) and meshless local Petrov–Galerkin (MLPG) methods. Also, it is not needed to determine shape parameter which is often played important role in these methods, for instance recall test (weight functions) for moving least squares (MLS) approximation in the frame of MLPG. Therefore, computational costs of SMRPI method is less expensive. A comparison study of the efficiency and accuracy of the present method and meshless local Petrov–Galerkin (MLPG) method is given by applying on mentioned diffusion equation. Convergence studies in the numerical examples show that SMRPI method possesses reasonable rate of convergence.