Bounds on the dimensions of trivariate spline spaces
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Bounds on the dimensions of trivariate spline spaces Peter Alfeld · Larry L. Schumaker
Received: 4 November 2006 / Accepted: 26 April 2007 / Published online: 5 December 2007 © Springer Science + Business Media, LLC 2007
Abstract We derive upper and lower bounds on the dimensions of trivariate spline spaces defined on tetrahedral partitions. The results hold for general partitions, and for all degrees of smoothness r and polynomial degrees d. Keywords Dimension · Trivariate splines Mathematics Subject Classification (2000) 41A15 1 Introduction Suppose is a finite collection of nondegenerate tetrahedra such that if any two tetrahedra in intersect, then their intersection is exactly one vertex, one edge, or one face. Then we call a tetrahedral partition of the set , where is the union of all tetrahedra in . This definition of a tetrahedral partition allows to have holes and cavities as well as pinch pointswhere just two tetrahedra meet in a single point. Given 0 ≤ r ≤ d, let Pd be the d+3 -dimensional space of trivariate polynomials 3 of degree d. In this paper we study the space
Sdr () = {s ∈ Cr () : s|T ∈ Pd , for all T ∈ } of trivariate splines of smoothness r and degree d associated with . Here the condition s ∈ Cr () means that s belongs to Cr (v) for every point v ∈ , i.e., if v
Communicated by Juan Manuel Peña. P. Alfeld (B) Department of Mathematics, University of Utah, 155 South 1400 East, JWB 233, Salt Lake City, UT 84112-0090, USA e-mail: [email protected] L. L. Schumaker Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA e-mail: [email protected]
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P. Alfeld, L.L. Schumaker
is contained in the tetrahedra T1 , . . . , Tm , then all of the polynomials s|T1 , . . . , s|Tm have common derivatives up to order r at v. Trivariate splines are important tools in approximation theory and numerical analysis, and have attracted considerable interest in the past few years, see [3] and references therein. Clearly Sdr () is a finite dimensional linear space. However, finding explicit formulae for its dimension for general r, d, and , is an extremely difficult problem due to the fact that the dimension depends not only on the way in which the tetrahedra are connected to each other, but also on the precise location of the vertices. In general, an arbitrarily small change of the location of the vertices can change the dimension of Sdr (). The dimension of Sdr () can be trivially determined when r = 0 or when r = d, see Remark 1. Our aim in this paper is to establish upper and lower bounds on the dimension of Sdr () for all choices of 0 < r < d, and for arbitrary tetrahedral partitions. Our bounds will depend on d and r, and also on the nature of , but not on the precise locations of the vertices. The paper is organized as follows. In Section 2 we describe how completely general tetrahedral partitions can be built inductively using 28 specific types of assembly steps. In Section 3 we show that lower and upper bounds on the dimension of Sdr () for arbitr
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