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$$n$$-widths, sup-infs, and optimality ratios for the $$k$$-version of the isogeometric finite element method. (English) Zbl 1227.65093
Summary: 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.

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
EIGIFP
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