Dimension of polynomial splines of mixed smoothness on T-meshes

TL;DR

This paper investigates the dimension of polynomial splines with mixed smoothness on T-meshes, providing conditions for smoothness reduction while maintaining stability, with applications to hierarchical and non-hierarchical T-meshes.

Contribution

It introduces constructive conditions for smoothness reduction in spline spaces on T-meshes, ensuring stable dimension and applicability to various mesh types.

Findings

Stable dimension for mixed smoothness spline spaces on hierarchical T-meshes.

Conditions for reducing smoothness across mesh edges without losing stability.

Applicability to both hierarchical and non-hierarchical T-meshes.

Abstract

In this paper we study the dimension of splines of mixed smoothness on axis-aligned T-meshes. This is the setting when different orders of smoothness are required across the edges of the mesh. Given a spline space whose dimension is independent of its T-mesh's geometric embedding, we present constructive and sufficient conditions that ensure that the smoothness across a subset of the mesh edges can be reduced while maintaining stability of the dimension. The conditions have a simple geometric interpretation. Examples are presented to show the applicability of the results on both hierarchical and non-hierarchical T-meshes. For hierarchal T-meshes it is shown that mixed smoothness spline spaces that contain the space of PHT-splines (Deng et al., 2008) always have stable dimension.