Andrea Bressan

Andrea Bressan, Espen Sande

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.

Andrea Bressan, Tom Lyche

This paper analyzes the approximation properties of spaces of piece-wise tensor product polynomials over box meshes with a focus on application to IsoGeometric Analysis (IGA). The errors are measured in Lebesgue norms. Estimates of different types are considered: local and global, with full or reduced Sobolev seminorms. Attention is also paid to the dependence on the degree and exponential convergence is proved for the approximation of analytic functions.