n-Widths, sup–infs, and optimality ratios for the k-version of the isogeometric finite element method
n-Widths, sup–infs, and optimality ratios for the k-version of the isogeometric finite element method
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DOI：
10.1016/j.cma.2009.01.021
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发表时间：
2009-05
影响因子：
7.2
通讯作者：
John A. Evans;Y. Bazilevs;I. Babuska;T. Hughes
中科院分区：
文献类型：
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作者：
John A. Evans;Y. Bazilevs;I. Babuska;T. Hughes
利用Kolmogorov n宽度理论，我们开始了k方法的数学研究。K-方法是一种有限元技术，其中使用了高阶连续的样条基函数。这是等距分析这一新领域的一个基本特征。已有的研究表明，在结构动力学、波传播和湍流等应用领域，k-方法比经典有限元方法具有更多的优势。Kolmogorov n-Width和sup-inf被引入作为评估逼近函数有效性的工具。在本文中，我们用这些工具研究了k-方法的逼近性质。在回顾了理论结果之后，我们进行了数值研究，其中我们计算了一些一维情形的n-宽度和超信息函数。这一研究进一步揭示了k-方法的逼近性质。最后，我们对k-方法和经典有限元方法进行了比较研究，并分析了多项式逼近的稳健性。
We begin the mathematical study of the k-method utilizing the theory of Kolmogorov n-widths. The k-method is a finite element technique where spline basis functions of higher-order continuity are employed. It is a fundamental feature of the new field of isogeometric analysis. In previous works, it has been shown that using the k-method has many advantages over the classical finite element method in application areas such as structural dynamics, wave propagation, and turbulence. The Kolmogorov n-width and sup–inf were introduced as tools to assess the effectiveness of approximating functions. In this paper, we investigate the approximation properties of the k-method with these tools. Following a review of theoretical results, we conduct a numerical study in which we compute the n-width and sup–inf for a number of one-dimensional cases. This study sheds further light on the approximation properties of the k-method. We finish this paper with a comparison study of the k-method and the classical finite element method and an analysis of the robustness of polynomial approximation.