Approximation in FEM, DG and IGA: A Theoretical Comparison
This provides a theoretical justification for using smoother splines in isogeometric analysis over finite element and discontinuous Galerkin methods, but the result is incremental as it confirms known practical observations.
The paper proves that for a given space dimension, C^{p-1} splines provide better a priori error bounds than C^{-1} (discontinuous) splines for approximating functions in H^{p+1}(0,1), and for C^0 splines the result holds for p≥3 but not for p=2. The results extend to broken Sobolev and tensor product spaces.
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.