On the dimension of spline spaces on planar T-meshes

Mourrain, Bernard

doi:10.1090/S0025-5718-2013-02738-X

On the dimension of spline spaces on planar T-meshes
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by Bernard Mourrain;

Math. Comp. 83 (2014), 847-871

DOI: https://doi.org/10.1090/S0025-5718-2013-02738-X

Published electronically: July 12, 2013

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Abstract:

We analyze the space $\mathcal {S}_{m, m’}^{\mathbf {r}} (\mathcal {T})$ of bivariate functions that are piecewise polynomial of bi-degree $\leqslant (m, m’)$ and of smoothness $\mathbf {r}$ along the interior edges of a planar T-mesh $\mathcal {T}$. We give new combinatorial lower and upper bounds for the dimension of this space by exploiting homological techniques. We relate this dimension to the weight of the maximal interior segments of the T-mesh, defined for an ordering of these maximal interior segments. We show that the lower and upper bounds coincide, for high enough degrees or for hierarchical T-meshes which are regular enough. We give a rule of subdivision to construct hierarchical T-meshes for which these lower and upper bounds coincide. Finally, we illustrate these results by analyzing spline spaces of small degrees and smoothness.

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