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An expansion of bivariate spline functions

Published online by Cambridge University Press:  17 February 2009

Huan-Wen Liu
Huan-Wen Liu
Affiliation:
School of Mathematics and Applied Statistics, The University of Wollongong, Wollongong, NSW 2522, Australia.
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Abstract

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Let Δ denote a triangulation of a planar polygon Ω. For any positive integer 0 ≤ r < k, let denote the vector space of functions in Cr whose restrictions to each triangle of Δ are polynomials of total degree at most k. Such spaces, called bivariate spline spaces, have many applications in surface fitting, scattered data interpolation, function approximation and numerical solutions of partial differential equations. An important problem is to give the function expression. In this paper, we prove that, if (Δ, Ω) is type-X, then any bivariate spline function in can be expressed by a series of univariate polynomials and a special bivariate finite element function in satisfying a so-called integral conformality condition system. We also give a direct sum decomposition of the space . In addition, the dimension of for a kind of triangulation has been determined.

Type
Research Article
Information
The ANZIAM Journal , Volume 41 , Issue 4 , April 2000 , pp. 527 - 541
Copyright
Copyright © Australian Mathematical Society 2000

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