A characterization of supersmoothness of multivariate splines
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A characterization of supersmoothness of multivariate splines Michael S. Floater1 · Kaibo Hu2 Received: 2 July 2019 / Accepted: 6 August 2020 / © The Author(s) 2020
Abstract We consider spline functions over simplicial meshes in Rn . We assume that the spline pieces join together with some finite order of smoothness but the pieces themselves are infinitely smooth. Such splines can have extra orders of smoothness at a vertex, a property known as supersmoothness, which plays a role in the construction of multivariate splines and in the finite element method. In this paper, we characterize supersmoothness in terms of the degeneracy of spaces of polynomial splines over the cell of simplices sharing the vertex, and use it to determine the maximal order of supersmoothness of various cell configurations. Keywords Supersmoothness · Spline · Finite element · Macroelement Mathematics Subject Classification (2010) Primary: 41A15 · 65D07 · Secondary: 41A58 · 65N30
1 Introduction Polynomial splines over a simplicial partition of a domain in Rn (a triangular mesh in 2D, a tetrahedral mesh in 3D, and so on) are functions whose pieces are polynomials Communicated by:Larry L. Schumaker KH was supported in part by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 339643, during his affiliation with the University of Oslo. Michael S. Floater
[email protected] Kaibo Hu [email protected] 1
Department of Mathematics, University of Oslo, Moltke Moes vei 35, 0851, Oslo, Norway
2
School of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, MN, 55455-0488, USA
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Adv Comput Math
(2020) 46:70
up to a certain degree d and which join together with some order of continuity r. Such spline functions may have extra orders of smoothness at a vertex of the mesh, a property known as supersmoothness as suggested by Sorokina [14]. For example, the Clough-Tocher macroelement, which is C 1 piecewise cubic, is twice differentiable at the refinement point, as first observed by Farin [5], and so this element can be said to have supersmoothness of order 2 at that point. For the construction of splines or finite elements with higher orders of continuity, it is important to recognize and make use of supersmoothness. For example, it plays a role in many of the macroelement constructions surveyed by Lai and Schumaker [8], where applications of splines to approximation theory and computer-aided geometric design are discussed. The concept of supersmoothness is also relevant to the finite element method. Motivated by structure-preserving or compatible discretizations, there has recently been an increased interest in investigating the use of splines for vector fields and differential complexes [2–4, 7]. The de Rham complex reveals a connection between smooth, e.g., C 1 , finite elements and the Stokes problem in fluid mechanics. In a discrete de Rham complex, the spline spaces for the velocity field may inherit the supersmoothness of the scalar field [2,
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