On the stability of the dimensions of spline spaces with highest order of smoothness over T-meshes

Bingru Huang, Falai Chen

Abstract: This paper studies the stability of the dimension of the spline space Sd(T) of bi-degree (d,d) with highest order of smoothness over a T-mesh T. By decomposing the T-connected component of T into a diagonalizable component and a non-diagonalizable component, we prove that the stability of the dimension of the spline space Sd(T) depends only on the stability of the rank of a matrix M̃ corresponding to the multi-vertices of the non-diagonalizable component. The matrix M̃ has a much smaller size than the conformality matrix associated with the T-connected component, and thus the stability is much easier to verify. As an application, we reprove the stability of the dimension of the spline space S3(T) over a T-mesh which is generated by subdividing a collection of 2 × 2 submeshes of a tensor product mesh under cross subdivision.