文档介绍：广西民族大学
硕士学位论文
非收缩三角剖分下二元样条函数空间的维数
姓名:衣娜
申请学位级别:硕士
专业:计算数学
指导教师:刘焕文
20080401
非收缩三角剖分下二元样条函数空间的维数
摘要
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平面上正规三角剖分下的二元样条函数空间 S n(∆) 在众多领域都有着广泛
的应用, 因而其维数问题受到了数学和计算机领域学者们的持续关注. 但人们很
早就发现其维数不仅依赖于三角剖分的拓扑性质, 而且强烈地依赖于剖分的几
何形状, 使得维数的确定非常困难. 对于一般正规三角剖分, 维数问题得到解决
的情形是 n ≥ 3r + 2 以及 n = 4, r = 1.
本学位论文主要研究非收缩三角剖分与广义非收缩三角剖分下, 样条函数
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空间 S n(∆) 的维数问题. 首先通过讨论星型域上二元样条空间的最小决定集的分
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布情况得到了一类特殊三角剖分–非收缩三角剖分下的二元样条函数空间 S n(∆)
的维数; 然后引入了两个构造三角剖分的新算子, 定义了一类广义非收缩三角剖
分, 它是对 Farin(2006) 提出的非收缩三角剖分的一个推广, 其中包含了一些众所
周知的难以在其上讨论维数的剖分, 如 Morgan-Scott 剖分, Robbins 剖分等. 并讨论
了广义非收缩三角剖分下二元三次一阶光滑样条函数空间的维数问题.
关键词:二元样条函数空间 B 网方法非收缩三角剖分广义非收缩三角剖
分最小决定集维数递推方法
DIMENSIONS OF BIVARIATE SPLINE SPACES OVER
UNCONSTRICTED TRIANGULATIONS
ABSTRACT
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The bivariate spline spaces S n(∆) on regular triangulations have very pop-
ular use in many fields. Therefore, its dimension problem has been continuously
concerned by the mathematics puter scholars. However, it is found that
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the dimensions of S n(∆) depend not only on the topological properties of trian-
gulations, but also on the geometric shape. This makes it difficult to determine
the dimensions. For the general regular triangulations, when n ≥ 3r + 2 and
n = 4, r = 1, the problem of dimensions has been solved.
In this paper, we focus on the problem of dimensions of spline function
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space S n(∆) under the unconstricted triangulations and generalized unconstricted
triangulations. The contents are arranged as follows. First, the dimensions of
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bivariate spline function spaces S n(∆) under a class of special triangulations –
unconstricted triangulations are achieved by studying the distribution of minimal
determining set on star-type regional. Then, by introducing two new operators
in triangulation construction, a kind of generalized unconstricted triangulation is
defined, which is an expansion of the unconstricted triangulations introduced by
Farin in 2006. It contains some well-known triangulations, such