Stable Local Bases for Multivariate Spline Spaces

Davydov, Oleg

doi:10.1006/jath.2001.3577

Cited by 17 publications

(22 citation statements)

References 23 publications

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“…By using the explicit dual basis (2.1) and the locality of the supports of s i,k (2.2), it is easy to show (see, e.g., Lemma 6.2 in [14]) that B FE is a Riesz basis for S 1 3 ( ), i.e., for any square summable coefficient vector c, it is true that…”

Section: Finite-element Basismentioning

confidence: 99%

Cubic Spline Prewavelets on the Four-Directional Mesh

Buhmann

Davydov²,

Goodman

2003

Foundations of Computational Mathematics

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Dedicated to Professor M. J. D. Powell on the occasion of his sixty-fifth birthday and his retirement.Abstract. In this paper, we design differentiable, two-dimensional, piecewise polynomial cubic prewavelets of particularly small compact support. They are given in closed form, and provide stable, orthogonal decompositions of L 2 (R 2 ). In particular, the splines we use in our prewavelet constructions give rise to stable bases of spline spaces that contain all cubic polynomials, whereas the more familiar box spline constructions cannot reproduce all cubic polynomials, unless resorting to a box spline of higher polynomial degree.

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Section: Finite-element Basismentioning

confidence: 99%

Cubic Spline Prewavelets on the Four-Directional Mesh

Buhmann

Davydov²,

Goodman

2003

Foundations of Computational Mathematics

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“…Surprisingly, without giving dimension formulae, it is possible to show that the spaces S r d (△) have local bases for d ≥ 8r + 1, see [5,7,8]. Numerical methods for constructing stable local bases of multivariate spline spaces can be found in [11].…”

Section: Remarkmentioning

confidence: 99%

“…[1][2][3][4][5][6][7][8][11][12][13][14][15]17,18,21,22]. For convenience we review the notation and basic concepts.…”

Section: §1 Introductionmentioning

confidence: 99%

Trivariate Cr Polynomial Macroelements

Lai

Schumaker

2006

Constr Approx

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Abstract. Trivariate C r macro-elements defined in terms of polynomials of degree 8r + 1 on tetrahedra are analyzed. For r = 1, 2, these spaces reduce to well-known macro-element spaces used in data fitting and in the finite-element method. We determine the dimension of these spaces, and describe stable local minimal determining sets and nodal minimal determining sets. We also show that the spaces approximate smooth functions to optimal order. §1. Introduction Let △ be a tetrahedral partition of a set Ω ∈ IR 3 . We denote the sets of vertices, edges, and faces of △, by V, E, and F , respectively. In this paper we study the superspline spaces ∈ C 2r (e), all e ∈ E},where in general we write P d for the d+3 3 dimensional space of trivariate polynomials of degree d. Here s ∈ C ρ (v) means that all polynomial pieces s| T associated with tetrahedra T sharing the vertex v have common derivatives up to order ρ at v. Similarly, s ∈ C µ (e) means that all polynomial pieces s| T associated with tetrahedra T sharing the edge e have common derivatives up to order µ at all points along the edge e.For r = 1 this space corresponds to a macro-element space first introduced in the finite-element literature in [23]. The analogous C 2 macro-element was developed in [16]. Both authors described their elements in terms of Hermite interpolation. It is well known, see Remark 1, that in order to construct similar macro-element spaces for higher values of r, we must work with splines of degree 8r + 1, and we must enforce C 4r supersmoothness at the vertices and C 2r supersmoothness around the edges of △. This suggest a natural set of Hermite data to associate with the element. But it is a nontrivial problem to describe what additional data 1)

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“…However, these bases are stable only for triangulations satisfying (in R 2 ) the minimal angle condition. We extend the construction of [22] to a wider class of strong locally regular triangulations, see §2 for a definition. Note that the new basis functions are invariant under affine transforms (see Remark 4.9).…”

Section: Introductionmentioning

confidence: 99%

“…However, all other arguments of our article are basically "dimension independent", and we refrain here from treating the case d > 2 only for the sake of simplicity and clarity. Therefore, we build upon the nodal spline bases of [22], which is the only known approach that produces stable local bases for nested spline spaces on general triangulations in all dimensions. However, these bases are stable only for triangulations satisfying (in R 2 ) the minimal angle condition.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Approximation from Differentiable Piecewise Polynomials

Davydov¹,

Petrushev²

2003

SIAM J. Math. Anal.

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We study nonlinear n-term approximation in L p (R 2 ) (0 < p ≤ ∞) from hierarchical sequences of stable local bases consisting of differentiable (i.e., C r with r ≥ 1) piecewise polynomials (splines). We construct such sequences of bases over multilevel nested triangulations of R 2 , which allow arbitrarily sharp angles. To quantize nonlinear nterm spline approximation, we introduce and explore a collection of smoothness spaces (B-spaces). We utilize the B-spaces to prove companion Jackson and Bernstein estimates and then characterize the rates of approximation by interpolation. Even when applied on uniform triangulations with well-known families of basis functions such as box splines, these results give a more complete characterization of the approximation rates than the existing ones involving Besov spaces. Our results can easily be extended to properly defined multilevel triangulations in R d , d > 2.

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Stable Local Bases for Multivariate Spline Spaces

Abstract: We present an algorithm for constructing stable local bases for the spaces S r d (2) of multivariate polynomial splines of smoothness r 1 and degree d r2 n +1 on an arbitrary triangulation 2 of a bounded polyhedral domain 0/R n , n 2. 2001Academic Press

Cited by 17 publications

References 23 publications

Cubic Spline Prewavelets on the Four-Directional Mesh

Cubic Spline Prewavelets on the Four-Directional Mesh

Trivariate Cr Polynomial Macroelements

Nonlinear Approximation from Differentiable Piecewise Polynomials

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