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.

翻译：本文分析了从样条函数中移除任意节点所产生的误差。当节点重数大于1时，这相当于将其重数减少一个单位。特别地，我们推导出一个非常简单的公式，通过若干相邻节点及所考虑样条的少数控制点来计算该误差。此外，我们精确展示了该误差如何与样条在节点处导数的跳跃相关。随后，我们利用所发展的理论，提出了高效且成本极低的局部误差指示器及自适应粗化算法。最后，我们通过数值实验展示其性能，并给出一些应用实例。