Abstract
We show that a nested sequence of C r macro-element spline spaces on quasi-uniform triangulations gives rise to hierarchical Riesz bases of Sobolev spaces H s(Ω), \(1<s<r+\frac{3}{2}\), and \(H^s_0(\Omega)\), \(1<s<\sigma+\frac{3}{2}\), \(s\notin\mathbb{Z}+\frac{1}{2}\), as soon as there is a nested sequence of Lagrange interpolation sets with uniformly local and bounded basis functions, and, in case of \(H^s_0(\Omega)\), the nodal interpolation operators associated with the macro-element spaces are boundary conforming of order σ. In addition, we provide a brief review of the existing constructions of C 1 Largange type hierarchical bases.
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Davydov, O., Yeo, W.P. (2014). Macro-element Hierarchical Riesz Bases. In: Floater, M., Lyche, T., Mazure, ML., Mørken, K., Schumaker, L.L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2012. Lecture Notes in Computer Science, vol 8177. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54382-1_7
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DOI: https://doi.org/10.1007/978-3-642-54382-1_7
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