The Instability in the Dimensions of Polynomial Splines of Mixed Smoothness over T-Meshes

Mixed-smoothness splines facilitate localized control over smoothness; however, the issue of dimensional instability in mixed-smoothness spline spaces remains unstudied in the existing literature. This paper studies such instabilities over T-meshes, where different orders of smoothness are required across interior mesh segments. Using the smoothing cofactor-conformality method, we introduce a constraint on T-meshes to derive a stable dimension formula for mixed-smoothness spline spaces. Furthermore, we show dimensional instability in cases involving T-cycles and nested T-cycles. By defining a singularity factor for each T-cycle, we demonstrate that both dimensional instabilities and structural degenerations are associated with these singularity factors. The work contributes to a deeper understanding of spline spaces defined over non-tensor-product structures.