\(n\)-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.
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 |
Keywords:
finite element methods; isogeometric analysis; \(k\)-method; approximation; \(n\)-widths; sup; infSoftware:
EIGIFPReferences:
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