Stable Simplex Spline Bases for \(C^3\) Quintics on the Powell–Sabin 12-Split

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• 作者：Tom Lyche ; Georg Muntingh
• 关键词：Stable basis ; Powell–Sabin 12 ; Split ; Simplex splines ; Marsden identity ; Quasi ; interpolation
• 刊名：Constructive Approximation
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：45
• 期：1
• 页码：1-32
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Numerical Analysis; Analysis;
• 出版者：Springer US
• ISSN：1432-0940
• 卷排序：45

文摘

For the space of \(C^3\) quintics on the Powell–Sabin 12-split of a triangle, we determine explicitly the six symmetric simplex spline bases that reduce to a B-spline basis on each edge and have a positive partition of unity, a Marsden identity that splits into real linear factors, and an intuitive domain mesh. The bases are stable in the \(L_\infty \) norm with a condition number independent of the geometry and have a well-conditioned Lagrange interpolant at the domain points and a quasi-interpolant with local approximation order 6. We show an \(h^2\) bound for the distance between the control points and the values of a spline at the corresponding domain points. For one of these bases, we derive \(C^0\), \(C^1\), \(C^2\), and \(C^3\) conditions on the control points of two splines on adjacent macrotriangles.