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Title: Refinable stable local bivariate spline bases on Powell-Sabin-12 triangulations
Author: Yeo, Wee Ping
Awarding Body: University of Strathclyde
Current Institution: University of Strathclyde
Date of Award: 2013
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In this thesis, we investigate the hierarchical bases of C¹ quadratic spline functions on Powell-Sabin-12 triangulations and their applications to surface compression and numerical solutions of the biharmonic equation. We also construct C² quintic refinable spaces of spline functions on the Powell-Sabin-12 triangulations. We first show that a nested sequence of Cr macro-element spline spaces on quasi-uniform triangulations gives rise to hierarchical Riesz bases of Sobolev spaces Hs(Ω), 1 < s < r + 3/2 , and Hs0(Ω), 1 < s < δ + 3/2, s /ε Z + 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 Hs0(Ω), the nodal interpolation operators associated with the macro-element spaces are boundary conforming of order s = r. Starting with a nested sequence of C¹ piecewise quadratic macro-element spaces which is generated by uniform refinements and combined Powell-Sabin-6 and -12 splits on arbitrary polygonal domains Ω C R² we construct hierarchical bases of Lagrange type. Properly normalised, these bases are Riesz bases for Sobolev spaces Hs(Ω), with s ε (1, 5/2) and Hs 0(Ω) for s ε (1, 3/2) U (3/2, 5/2). Compared to the previously constructed C¹ Lagrange hierarchical bases of [21, 41] which require some special partitions of the initial triangulations, our bases are constructed on general triangulations. Our bases have larger stability range for Hs(Ω) compared to the C¹ wavelet bases of [33] and the C¹ Hermite type hierarchical bases of [14, 49]. Numerical results are presented to show the advantages of our Lagrange hierarchical basis in compressing surfaces. Since this basis is a Riesz basis for the space H2 0 (Ω), we also investigate the use of the hierarchical basis as a preconditioner for solving the biharmonic equation. In addition, we propose a construction of refinable spaces of C² macro-elements of degree 5 on triangulations of a polygonal domain obtained by uniform refinements of an initial triangulation and a Powell-Sabin-12 split. The new refinable macro-elements have stable local minimal determining sets (MDS). Therefore they can be used to construct nested spline spaces which possess stable local bases and also achieve optimal approximation power.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available