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A parsimonious approach to $C^2$ cubic splines on arbitrary triangulations: Reduced macro-elements on the cubic Wang-Shi split

Tom Lyche, Carla Manni, Hendrik Speleers

TL;DR

This paper addresses constructing $C^2$ cubic splines on the $WS_3$ refinement of an arbitrary triangulation by concentrating Hermite degrees of freedom at vertices and edges, achieving a parsimonious representation via macro-elements. It develops reduced subspaces ${\mathbb S}_3^{2,m}({\mathcal T}_{WS_3})$ containing ${\mathbb P}_3$ and represented by a local simplex-spline basis, enabling per-triangle construction and straightforward global coupling. The approach yields subspaces with dimension up to six times the number of vertices and provides explicit conditions and procedures to preserve cubic polynomial reproduction while removing interior degrees of freedom. The resulting framework offers an efficient and geometrically robust method for $C^2$ cubic splines on complex triangulations, with practical implementation aided by the simplex-spline basis and the necessary basis-conversion matrices.

Abstract

We present a general method to obtain interesting subspaces of the $C^2$ cubic spline space defined on the cubic Wang-Shi refinement of a given arbitrary triangulation $\mathcal{T}$. These subspaces are characterized by specific Hermite degrees of freedom associated with only the vertices and edges of $\mathcal{T}$, or even only the vertices of $\mathcal{T}$. Each subspace still contains cubic polynomials while saving a consistent number of degrees of freedom compared with the full space. The dimension of the considered subspaces can be as small as six times the number of vertices of $\mathcal{T}$. The method fits in the setting of macro-elements: any function of such a subspace can be constructed on each triangle of $\mathcal{T}$ separately by specifying the necessary Hermite degrees of freedom. The explicit local representation in terms of a local simplex spline basis is also provided. This simplex spline basis intrinsically takes care of the complex geometry of the Wang-Shi split, making it transparent to the user.

A parsimonious approach to $C^2$ cubic splines on arbitrary triangulations: Reduced macro-elements on the cubic Wang-Shi split

TL;DR

This paper addresses constructing cubic splines on the refinement of an arbitrary triangulation by concentrating Hermite degrees of freedom at vertices and edges, achieving a parsimonious representation via macro-elements. It develops reduced subspaces containing and represented by a local simplex-spline basis, enabling per-triangle construction and straightforward global coupling. The approach yields subspaces with dimension up to six times the number of vertices and provides explicit conditions and procedures to preserve cubic polynomial reproduction while removing interior degrees of freedom. The resulting framework offers an efficient and geometrically robust method for cubic splines on complex triangulations, with practical implementation aided by the simplex-spline basis and the necessary basis-conversion matrices.

Abstract

We present a general method to obtain interesting subspaces of the cubic spline space defined on the cubic Wang-Shi refinement of a given arbitrary triangulation . These subspaces are characterized by specific Hermite degrees of freedom associated with only the vertices and edges of , or even only the vertices of . Each subspace still contains cubic polynomials while saving a consistent number of degrees of freedom compared with the full space. The dimension of the considered subspaces can be as small as six times the number of vertices of . The method fits in the setting of macro-elements: any function of such a subspace can be constructed on each triangle of separately by specifying the necessary Hermite degrees of freedom. The explicit local representation in terms of a local simplex spline basis is also provided. This simplex spline basis intrinsically takes care of the complex geometry of the Wang-Shi split, making it transparent to the user.

Paper Structure

This paper contains 12 sections, 3 theorems, 53 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The cubic spaces ${\mathbb{S}}_3^2(\Delta_{{{\rm WS}_3}})$ and ${\mathbb{S}}_3^2({\mathcal{T}}_{{{\rm WS}_3}})$
  3. Definition of the spaces
  4. Local simplex spline basis for ${\mathbb{S}}_3^2(\Delta_{{{\rm WS}_3}})$
  5. Reduced spaces
  6. Reduced spaces of ${\mathbb{S}}_3^2(\Delta_{{\rm WS}_3})$
  7. Peeling away Hermite degrees of freedom
  8. Hermite and simplex spline representation of subspaces of ${\mathbb{S}}_3^2(\Delta_{{{\rm WS}_3}})$
  9. Subspaces of ${\mathbb{S}}_3^2(\Delta_{{{\rm WS}_3}})$ containing cubic polynomials
  10. Removing the interior degree of freedom
  11. Reduced spaces of ${\mathbb{S}}_3^2({\mathcal{T}}_{{\rm WS}_3})$
  12. Conclusion

Key Result

Proposition 1

For given data $f_{k,\alpha,\beta}$, $g_k$, $g_{k,l}$, and $h_0$, there exists a unique spline $s\in{\mathbb{S}}_3^2(\Delta_{{{\rm WS}_3}})$ such that where ${\boldsymbol{n}}_k$ is the normal direction of the edge opposite to vertex ${\boldsymbol{p}}_k$, and the points ${\boldsymbol{p}}_{k,l}$, ${\boldsymbol{q}}_{k}$, and ${\boldsymbol{q}}$ are defined in eq:knot-points and eq:dof-points-1.

Figures (6)

  • Figure 1: ${{\rm WS}_d}$ splits for $d=2,3,4$.
  • Figure 2: Labeling of the knots on the boundary of the triangle $\Delta$.
  • Figure 3: Hermite degrees of freedom on the ${{\rm WS}_3}$ split.
  • Figure 4: Knot configurations of the considered simplex spline basis functions for ${\mathbb{S}}_3^2(\Delta_{{{\rm WS}_3}})$. Each black disc shows the position of a knot and the number inside indicates its multiplicity.
  • Figure 5: The simplex spline basis functions $B_1$, $B_4$, and $B_{10}$.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Proposition 1
  • Remark 2
  • Example 3
  • Proposition 4
  • proof
  • Remark 5
  • Proposition 6
  • proof
  • Remark 7
  • Remark 8
  • ...and 3 more