Module LRSplines2D

LR-mesh

Box. Given an integer $d\geq 1$, a box in $\mathbb{R}^d$ is a Cartesian product $\beta = J_1 \times \cdots \times J_d \subset \mathbb{R}^d$, where each $J_k = [a_k, b_k]$ with $a_k\leq b_k$ is a closed finite interval in $\mathbb{R}$. We write $\dim \beta$ for the dimension of $\beta$, defined as the number of intervals $J_k$ for which $a_k\neq b_k$. We call $\beta$ an $\ell$-box if $\dim \beta = \ell$.

If $\dim \beta = d$ then $\beta$ is called an element, while if $\dim \beta = d-1$, then there exists exactly one $k$ such that $J_k$ has a single element, i.e., the box has zero thickness in the $k$-th direction. In this case $\beta$ is called a mesh-rectangle or a $k$-mesh-rectangle.

In order to consider splines on the domain $\Omega$ with different orders of continuity across different mesh-rectangles, it is natural to assign to each mesh-rectangle a multiplicity, encoded as a function $\mu: \mathcal{M}\longrightarrow \mathbb{N}$. The corresponding pair $(\mathcal{M}, \mu)$ is called a $\mu$-extended box-mesh.

Mesh. Let $\Omega \subset \mathbb{R}^d$ be a $d$-box in $\mathbb{R}^d$. A box partition $\mathcal{E} = \{\beta_1, \ldots, \beta_n\}$ of $\Omega$ is a finite collection of $d$-boxes in $\mathbb{R}^d$ such that $\beta_1\cup \cdots\cup \beta_n = \Omega$ and $\mathring{\beta}_i \cap \mathring{\beta}_j = \emptyset$ whenever $i\neq j$.

Tensor-mesh. A special case is the tensor-mesh $\mathcal{E} = \{[a_{1,i_1},a_{1,i_1+1}]\times\cdots\times[a_{d,i_d},a_{d,i_d+1}] : 1\leq i_k\leq n_k - 1, k = 1,\ldots, d\}$, for given sequences $a_{k,1} < \cdots < a_{k,n_k}$, where $k=1,\ldots,d$.

LR-mesh. LR-meshes are constructed by successive refinement of box-meshes such that in each refinement at least one $d$-box is split in two by a mesh-rectangle $\gamma$, or by increasing multiplicity as below.

Splits. Given a mesh-rectangle $\gamma$ and a $d$-box $\beta$ in $\mathbb{R}^d$, we say that $\gamma$ splits $\beta$ if $\beta \backslash \gamma$ is not connected. Moreover, $\gamma$ is a minimal split of $\beta$ if it splits $\beta$ and $\gamma \subset \beta$. Let be given a box partition $\mathcal{E}$ of a $d$-box $\Omega$ and a mesh-rectangle $\gamma$ in $\mathbb{R}^d$. Then $\gamma$ splits $\mathcal{E}$ if $\gamma$ is a finite union $\bigcup_i \gamma_i$ of mesh-rectangles such that each $\gamma_i$ is either a minimal split of a box in $\mathcal{E}$, or is an existing mesh-rectangle in $\mathcal{M}(\mathcal{E})$.

• A $k$-constant split is a union $\gamma = \bigcup_i \gamma_i$ of $k$-meshrectangles $\gamma_i$ ...
• A ($\mu$-extended) LR-mesh is a $\mu$-extended box-mesh $(\mathcal{M}, \mu)$, recursively defined as either
  • $(\mathcal{M}, \mu)$ is a $\mu$-extended tensor-mesh, or
  • $(\mathcal{M}, \mu) = (\overline{\mathcal{M}} \cup \gamma, \overline{\mu})$, where $(\overline{\mathcal{M}}, \overline{\mu})$ is a $\mu$-extended LR-mesh and $\gamma$ is a constant split of $(\overline{\mathcal{M}}, \overline{\mu})$.

Sources

• T. Dokken, T. Lyche, K.F. Pettersen. Polynomial splines over locally refined box-partitions, Computer Aided Geometric Design, Volume 30, Issue 3, March 2013, Pages 331--356