A note on the conditioning of a class of generalized finite element methods

We study the asymptotic behavior of the condition number of the linear system from the discretization of a class of generalized finite element methods for solving second-order elliptic boundary value problems. Allowing local approximation spaces with polynomials of different degrees and different local patch sizes (local refinements), we give bounds on the condition number in relation to the patch size and the dimension of the global approximation space in which the shape functions are in general not polynomials. Numerical tests verify the theorems.