Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis

TL;DR

This paper derives explicit a priori error estimates for spline approximation in isogeometric analysis, demonstrating that smoother spline spaces improve approximation quality, with results applicable to univariate, multivariate, and mapped geometries.

Contribution

It extends existing error estimates to spline spaces of arbitrary smoothness and arbitrary grids, including multi-patch geometries, with explicit constants.

Findings

Smoother spline spaces yield better approximation per degree of freedom.

Error estimates are validated with numerical evidence.

Results apply to both univariate and multivariate tensor-product spline spaces.

Abstract

In this paper we provide a priori error estimates with explicit constants for both the $L^2$-projection and the Ritz projection onto spline spaces of arbitrary smoothness defined on arbitrary grids. This extends the results recently obtained for spline spaces of maximal smoothness. The presented error estimates are in agreement with the numerical evidence found in the literature that smoother spline spaces exhibit a better approximation behavior per degree of freedom, even for low smoothness of the functions to be approximated. First we introduce results for univariate spline spaces, and then we address multivariate tensor-product spline spaces and isogeometric spline spaces generated by means of a mapped geometry, both in the single-patch and in the multi-patch case.