Approximation in FEM, DG and IGA: A Theoretical Comparison

TL;DR

This paper theoretically compares the approximation capabilities of spline spaces with varying degrees of smoothness in FEM, DG, and IGA, demonstrating that higher smoothness generally yields better error bounds.

Contribution

It provides a rigorous comparison of approximation properties of spline spaces with different continuity levels, establishing when higher smoothness improves approximation error bounds.

Findings

$oldsymbol{ ext{smooth } p-1}$ splines outperform discontinuous splines for a given space dimension.

Results are generalized to functions in different Sobolev spaces and tensor product spaces.

Theoretical bounds are established for approximation errors in various spline spaces.

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.