On the problem of instability in the dimension of a spline space over a T-mesh

In this paper, we discuss the problem of instability in the dimension of a spline space over a T-mesh. For bivariate spline spaces S(5,5,3,3) and S(4,4,2,2), the instability in the dimension is shown over certain types of T-meshes. This result could be considered as an attempt to answer the question of how large the polynomial degree (m,m′) should be relative to the smoothness (r,r′) to make the dimension of a spline space stable. We show in particular that the bound m≥2r+1 and m′≥2r′+1 are optimal. For certain types of T-meshes, we prove that the dimension of spaces S(4,4,2,2)(T), S(5,5,3,3)(T) depends on the coordinates of vertices of T-mesh T. We show, in particular case, that the bound m≥2r+1 and m′≥2r′+1 are optimal. Motivation: Splines over T-mesh could be a useful tool in many areas such as surface modeling and finite element analysis. The instability in the dimension of a space S(m,m′,r,r′) is an important issue to investigate, because it provides a better understanding of the main problem, namely, how the basis of a space S(m,m′,r,r′) could be described. [Display omitted] ► We examine the dimension of a bivariate spline space over T-mesh. ► We treat the case of polynomial degree 5, 5 and order of smoothness 3, 3. ► And we treat the case of polynomial degree 4, 4 and order of smoothness 2, 2. ► We provide the examples of T-meshes such the dimension depends not only on topology. ► But it also depends on the geometry of a given T-meshes.