  Generalized or extended finite element methods (GFEM/XFEM) are in general
badly conditioned and have numerous additional degrees of freedom (DOF)
compared with the FEM because of introduction of enriched functions. In this
paper, we develop an approach to establish a subspace of a conventional
GFEM/XFEM approximation space using partition of unity (PU) techniques and
local least square procedures. The proposed GFEM is referred to as condensed
GFEM (CGFEM), which (i) possesses as many DOFs as the preliminary FEM, (ii)
enjoys similar approximation properties with the GFEM/XFEM, and (iii) is
well-conditioned in a sense that its conditioning is of the same order as that
of the FEM. The fundamental approximation properties of CGFEM is proven
mathematically. The CGFEM is applied to a problem of high order polynomial
approximations and a Poisson crack problem; optimal convergence orders of the
former are proven rigorously. The numerical experiments and comparisons with
the conventional GFEM/XFEM and FEM are made to verify the theory and
effectiveness of CGFEM.
