Approximation in FEM, DG and IGA: A Theoretical Comparison

Abstract

In this paper we compare approximation properties of degree $p$ spline spaces with different numbers of continuous derivatives. We prove that, for a given space dimension, \smooth {p-1} splines provide better a priori error bounds for the approximation of functions in $H^{p+1}(0,1)$. Our result holds for all practically interesting cases when comparing \smooth {p-1} splines with \smooth {-1} (discontinuous) splines. When comparing \smooth {p-1} splines with \smooth 0 splines our proof covers almost all cases for $p\ge 3$, but we can not conclude anything for $p=2$. The results are generalized to the approximation of functions in $H^{q+1}(0,1)$ for $q<p$, to broken Sobolev spaces and to tensor product spaces.Comment: 21 pages, 4 figures. Fixed typos and improved the presentatio