Approximation properties of the generalized finite element method

In this paper, we have obtained an approximation result in the Gener-alized Finite Element Method (GFEM) that reflects the global approxima-tion property of the Partition of Unity (PU) as well as the approximability of the local approximation spaces. We have considered a GFEM, where the underlying PU functions reproduce polynomials of degree l. With the space of polynomials of degree k serving as the local approximation spaces of the GFEM, we have shown, in particular, that the energy norm of the GFEM approximation error of a smooth function is O(hl+k). Estimates in the W tp-norm have also been established. This result could not be ob-tained from the classical approximation result of GFEM, which does not reflect the global approximation property of the PU