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

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 GG 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 GG space theory can be ensured.