J. Comp. Math., 33 (2015), pp. 248-262. |
On the Problem of Instability in the Dimensions of Spline Spaces over T-Meshes with T-Cycles Qing-Jie Guo ^{1}, Ren-Hong Wang ^{1}, Chong-Jun Li ^{1} 1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, ChinaReceived 2013-8-7 Accepted 2014-11-17 Available online 2015-5-15 doi:10.4208/jcm.1411-m4419 Abstract The T-meshes are local modification of rectangular meshes which allow T-junctions. The splines over T-meshes are involved in many fields, such as finite element methods, CAGD etc. The dimension of a spline space is a basic problem for the theories and applications of splines. However, the problem of determining the dimension of a spline space is difficult since it heavily depends on the geometric properties of the partition. In many cases, the dimension is unstable. In this paper, we study the instability in the dimensions of spline spaces over T-meshes by using the smoothing cofactor-conformality method. The modified dimension formulas of spline spaces over T-meshes with T-cycles are also presented. Moreover, some examples are given to illustrate the instability in the dimensions of the spline spaces over some special meshes. Key words: Spline space, Smoothing cofactor-conformality method, Instability in the dimension, T-meshes. AMS subject classifications: 65D07. Email: guoqingjie@mail.dlut.edu.cn, renhong@dlut.edu.cn, chongjun@dlut.edu.cn |