Approximation in FEM, DG and IGA: a theoretical comparison

In this paper we compare the approximation properties of degree p spline spaces with different numbers of continuous derivatives. We prove that, for a given space dimension, Cp-1 splines provide better a priori error bounds for the approximation of functions in Hp+1(0, 1). Our result holds for all practically interesting cases when comparing Cp-1 splines with C-1 (discontinuous) splines. When comparing Cp-1 splines with C-0 splines our proof covers almost all cases for p >= 3, but we can not conclude anything for p = 2. The results are generalized to the approximation of functions in Hq+1(0, 1) for q < p, to broken Sobolev spaces and to tensor product spaces.