均匀二型剖分下的二元五次B样条基函数及其应用

1975年王仁宏建立了任意剖分下多元样条函数的基本理论框架,即所谓光滑余因子方法.多元样条在函数逼近、计算机辅助几何设计、有限元及小波等领域中均有重要的应用.由于某些特殊剖分如均匀剖分的可研究性,1984年王仁宏给出均匀二型剖分下的二元三次一阶光滑样条空间S13（Δm（2n））的维数及其B样条基函数,在计算机辅助几何设计,微分方程数值解等方面应用广泛.在研究光滑余因子方法的基础上,分析均匀二型剖分下的二元五次三阶光滑样条空间S35（Δm（2n））函数空间,给出了S35（Δm（2n））的维数及其B样条基函数,满足曲面拟合和微分方程数值解等应用中对更高阶光滑性的要求.基于该组基函数,提出一种Poisson方程的数值解方法,通过数值实例检验该方法的精度.

Numerical solution of Poisson equation by quintic B-spline interpolation

Multivariate splines have wide applications in approximation theory,computer aided geometric design(CAGD) and finite element method.In 1975,Ren-Hong Wang established a new approach to study the basic theory on multivariate spline functions on arbitrary partition by presenting the so called Smoothing cofactor-conformality method.As the large applications in CAGD et al.,Ren-Hong Wang discussed the dimension and B-spline basis of the C1 cubic spline spaces on type-2 triangulation partition,which is denoted by S13(Δ(2)mn).Accordingly we analyze the C3 quintic spline spaces on type-2 triangulation partition S35(Δ(2)mn).The dimension and one group of B spline basis of S35(Δ(2)mn)are given.High derivatives is satisfied in applications.Based on the basis one numerical scheme is proposed to simulate the Poisson equation.Numerical examples are given to show the validity of the scheme.