Thesis

Refinable stable local bivariate spline bases on Powell-Sabin-12 triangulations

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Awarding institution
  • University of Strathclyde
Date of award
  • 2013
Thesis identifier
  • T13647
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Department, School or Faculty
Abstract
  • 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 < σ + ³/₂, s /∈ Z + ½, 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 σ ≤ 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 ⊂ R² we construct hierarchical bases of Lagrange type. Properly normalised, these bases are Riesz bases for Sobolev spaces Hs(Ω), with s ∈ (1, ⁵/₂) and Hs 0(Ω) for s ∈ (1, 3/2) ∪ (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.
Resource Type
DOI
Date Created
  • 2013
Former identifier
  • 1004658

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