Compact moving least squares: An optimization framework for generating high-order compact meshless discretizations

A generalization of the optimization framework typically used in moving least squares is presented that provides high-order approximation while maintaining compact stencils and a consistent treatment of boundaries. The approach, which we refer to as compact moving least squares, resembles the capabilities of compact finite differences but requires no structure in the underlying set of nodes. An efficient collocation scheme is used to demonstrate the capabilities of the method to solve elliptic boundary value problems in strong form stably without the need for an expensive weak form. The flexibility of the approach is demonstrated by using the same framework to both solve a variety of elliptic problems and to generate implicit approximations to derivatives. Finally, an efficient preconditioner is presented for the steady Stokes equations, and the approach's efficiency and high order of accuracy is demonstrated for domains with curvi-linear boundaries.