Proofs of the stability and convergence of a weakened weak method using PIM shape functions

Abstract

Recently, the smoothed point interpolation method (S-PIM) regarded as a weakened weak (W2) formulation method has been developed for solving engineering mechanics problems. It works well with distorted meshes. The G space theory offers the theoretical base for all the W2 methods that use smoothing operations. In this paper, we first prove mathematically that if a function is of Lipschitz continuity and its interpolated function is established using PIM shape functions, then the interpolated function belongs to a Gh,0s space. Our proofs work for smoothing operations that are the node-based, cell-based and a mixture of both smoothing domains. In addition, when mesh is refined under a given regularity condition, a sufficiently smooth target function can be approximated by its interpolated function with arbitrary accuracy, meaning that the interpolation error norm approaches to zero. Therefore, the stability and convergence of a W2 method using PIM shape functions and G space theory can be ensured.