On Tensor-Product Bases of PHT-Spline Spaces

We show how to generate hierarchical T-meshes in \(\mathbb R^2\) with associated locally refined B-splines, which possess the property of local linear independence, form a non-negative partition of unity, and span the resulting spaces of \(C^s\) -smooth polynomial splines of degree \(p=2s+1\) . The bases are collections of systems of tensor-product B-splines, without any need for truncation or similar modifications. The construction extends our earlier results for \(s=0\) and the bilinear case  [12]. Additionally we introduce two new mesh quality parameters that control the local complexity and quality of the elements. We also analyze their impact on the resulting meshes with the help of numerous examples. In order to make the paper self-contained, we also include a new proof (covering the case \(p=2s+1\) ) of the fact – first noted by Dokken et al. [8] – that the resulting locally refined B-splines depend solely on the final mesh.