Dimensions of Biquadratic Spline Spaces over T-meshes

TL;DR

This paper analyzes the dimensions of biquadratic spline spaces over T-meshes, introducing new concepts and deriving formulas for various T-mesh structures, advancing understanding of spline space properties.

Contribution

It introduces the concepts of T-mesh extension and spline spaces with boundary conditions, and derives dimension formulas for biquadratic spline spaces over hierarchical and general T-meshes.

Findings

Dimension formula for bilinear spline spaces with zero smoothness.

Lower bound for biquadratic spline space dimensions.

Dimension formula for hierarchical T-meshes.

Abstract

This paper discusses the dimensions of the spline spaces over T-meshes with lower degree. Two new concepts are proposed: extension of T-meshes and spline spaces with homogeneous boundary conditions. In the dimension analysis, the key strategy is linear space embedding with the operator of mixed partial derivative. The dimension of the original space equals the difference between the dimension of the image space and the rank of the constraints which ensuring any element in the image space has a preimage in the original space. Then the dimension formula and basis function construction of bilinear spline spaces of smoothness order zero over T-meshes are discussed in detail, and a dimension lower bound of biquadratic spline spaces over general T-meshes is provided. Furthermore, using level structure of hierarchical T-meshes, a dimension formula of biquadratic spline space over hierarchical T-meshes are proved. A topological explantation of the dimension formula is shown as well.