n-Widths, sup–infs, and optimality ratios for the k-version of the isogeometric finite element method. Evans, J. A., Bazilevs, Y., Babuška, I., & Hughes, T. J. R. Computer Methods in Applied Mechanics and Engineering, 198(21-26):1726–1741, Elsevier, 2009. doi abstract bibtex 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.
@Article{ Evans_2009aa,
abstract = {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. },
author = {Evans, John A. and Bazilevs, Yuri and Babuška, Ivoo and Hughes, Thomas Joseph Robert},
doi = {http://dx.doi.org/10.1016/j.cma.2009.01.021},
file = {Evans_2009aa.pdf},
issn = {0045-7825},
journal = {Computer Methods in Applied Mechanics and Engineering},
keywords = {Sup–inf},
number = {21-26},
pages = {1726--1741},
publisher = {Elsevier},
title = {n-Widths, sup–infs, and optimality ratios for the k-version of the isogeometric finite element method},
volume = {198},
year = {2009},
shortjournal = {Comput. Meth. Appl. Mech. Eng.}
}
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