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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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.
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DOI
Date Created
  • 2013
Former identifier
  • 1004658

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