Error analysis for local coarsening in univariate spline spaces

TL;DR

This paper provides a detailed analysis of the error caused by local knot removal in univariate spline spaces, deriving formulas and error indicators to improve adaptive coarsening algorithms.

Contribution

It introduces a simple formula for error calculation and develops efficient local error indicators for adaptive spline coarsening.

Findings

Derived a simple error formula based on neighboring knots and control points.

Established a relationship between error and derivative jumps at knots.

Demonstrated the effectiveness of proposed algorithms through numerical experiments.

Abstract

In this article we analyze the error produced by the removal of an arbitrary knot from a spline function. When a knot has multiplicity greater than one, this implies a reduction of its multiplicity by one unit. In particular, we deduce a very simple formula to compute the error in terms of some neighboring knots and a few control points of the considered spline. Furthermore, we show precisely how this error is related to the jump of a derivative of the spline at the knot. We then use the developed theory to propose efficient and very low-cost local error indicators and adaptive coarsening algorithms. Finally, we present some numerical experiments to illustrate their performance and show some applications.