The piecewise polynomial partition of unity functions for the generalized finite element methods

A partition of unity (PU) function is an essential component of the generalized finite element method (GFEM). The popular Shepard PU functions, which are rational functions, are easy to construct, but have difficulties in dealing with essential boundary conditions and require lengthy computing time for reasonable accuracy in numerical integration. In this paper, we introduce two simple PU functions. The first is a highly regular piecewise polynomial consisting of two distinct polynomials that is effective for uniformly partitioned patches. The second is a highly regular piecewise polynomial consisting of three distinct polynomials which is for arbitrary partitioned patches.