Local Lagrange Interpolation with Bivariate Splines of Arbitrary Smoothness

Nürnberger, Günther; Rayevskaya, Vera; Schumaker, Larry L.; Zeilfelder, Frank

doi:10.1007/s00365-005-0600-2

Cited by 20 publications

(36 citation statements)

References 21 publications

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“…As a first step towards defining , we begin by sorting the cubes in ♦ into five different classes. This step is motivated by our earlier work on Lagrange interpolation with both bivariate and trivariate splines [18]- [19], [21]- [26].…”

Section: §2 Freudenthal Partitionsmentioning

confidence: 99%

“…Even the case of a uniform cube partition of the unit cube with n an even integer is more complicated than the odd case presented here. Bivariate results in [21], [25] can help guide the construction.…”

Section: Lemma 81 Fix F ∈ C(ω) and Let T Be A Tetrahedron In Supmentioning

confidence: 99%

“…For the bivariate case, the problem of constructing a Lagrange interpolating pair utilizing smooth splines defined on triangulations has been studied in a number of recent papers [18]- [19], [21]- [22], [24]- [26]. Our most recent paper [19] (see also [18]) deals with the general case of arbitrary smoothness and given initial triangulations. On the other hand, much less is known for the trivariate case; see [23], [29].…”

Section: Lemma 81 Fix F ∈ C(ω) and Let T Be A Tetrahedron In Supmentioning

confidence: 99%

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A local Lagrange interpolation method based on $C^{1}$ cubic splines on Freudenthal partitions

et al. 2007

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Abstract. A trivariate Lagrange interpolation method based on C 1 cubic splines is described. The splines are defined over a special refinement of the Freudenthal partition of a cube partition. The interpolating splines are uniquely determined by data values, but no derivatives are needed. The interpolation method is local and stable, provides optimal order approximation, and has linear complexity. §1. Introductionbe a set of points in R 3 . In this paper we are interested in the following problem. Problem 1. Find a tetrahedral partition whose set of vertices includes V, an Ndimensional space S of trivariate splines defined on with a prescribed smoothness r, and a set of additional points {η i } N i=n+1 such that for every choice of the data, there is a unique spline s ∈ S satisfying (1.1)We call P :and S a Lagrange interpolation pair. It is easy to solve this problem using C 0 splines; see Remark 1. In this case no additional interpolation points are needed. However, the situation is much more complicated if we want to use C r splines with r ≥ 1. For r = 1 the problem was first solved in [29] for the special case where the points in V are the vertices of a cube partition. The method uses quintic splines, and provides full approximation power six for sufficiently smooth functions. In [23] we recently described two C 1 methods which solve the problem for arbitrary initial point sets V. The first method is based on quadratic splines, and requires that each tetrahedron be split into 24 subtetrahedra. The second method is based on cubic splines, and requires that each tetrahedron be split into 12 subtetrahedra. Both methods provide approximation order three for sufficiently smooth functions. For the quadratic spline case this is the optimal approximation order, but for the cubic case it is suboptimal.The aim of this paper is to describe a new method based on cubic C 1 splines which yields optimal approximation order four. To achieve this, we restrict ourselves

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Section: §2 Freudenthal Partitionsmentioning

confidence: 99%

Section: Lemma 81 Fix F ∈ C(ω) and Let T Be A Tetrahedron In Supmentioning

confidence: 99%

Section: Lemma 81 Fix F ∈ C(ω) and Let T Be A Tetrahedron In Supmentioning

confidence: 99%

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A local Lagrange interpolation method based on $C^{1}$ cubic splines on Freudenthal partitions

et al. 2007

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“…A filter might take this set of points and introduce some type of continuity assumption to create the reconstructed solution f Ã ðxÞ. Filtering for visualization based upon discrete data is often constructed by considering the convolution of the "true" solution, of which f k is a sampling, against some type of spline, often a cubic B-spline [22], [23], [24], [25], [26], [27]. Much of the literature concentrates on the specific use in image processing, though there has also been work in graphic visualization [10], [28], as well as computer animation [29].…”

Section: Previous Work In Filtering For Visualizationmentioning

confidence: 99%

Investigation of Smoothness-Increasing Accuracy-Conserving Filters for Improving Streamline Integration through Discontinuous Fields

Steffen

Curtis

Kirby

et al. 2008

IEEE Trans. Visual. Comput. Graphics

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Abstract-Streamline integration of fields produced by computational fluid mechanics simulations is a commonly used tool for the investigation and analysis of fluid flow phenomena. Integration is often accomplished through the application of ordinary differential equation (ODE) integrators-integrators whose error characteristics are predicated on the smoothness of the field through which the streamline is being integrated, which is not available at the interelement level of finite volume and finite element data. Adaptive error control techniques are often used to ameliorate the challenge posed by interelement discontinuities. As the root of the difficulties is the discontinuous nature of the data, we present a complementary approach of applying smoothness-increasing accuracy-conserving filters to the data prior to streamline integration. We investigate whether such an approach applied to uniform quadrilateral discontinuous Galerkin (high-order finite volume) data can be used to augment current adaptive error control approaches. We discuss and demonstrate through a numerical example the computational trade-offs exhibited when one applies such a strategy.

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“…On domains that allow a conforming quadrangulation with all interior vertices of degree four, these triangulations are the standard triangulated quadrangulations (so called FVS triangulations) associated with the well known Fraeijs de Veubeke-Sander quadrilateral C 1 cubic finite elements. Recently, locally supported Lagrange bases for the spaces of C 1 piecewise cubic polynomials on FVS triangulations have appeared in [18,19,20].…”

Section: Introductionmentioning

confidence: 99%

Hierarchical Riesz Bases for Hs(Ω), 1 < s < 5/2

Davydov

Stevenson

2005

Constr Approx

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On arbitrary polygonal domains Ω ⊂ R 2 , we construct C 1 hierarchical Riesz bases for Sobolev spaces H s (Ω). In contrast to an earlier construction by Dahmen, Oswald and Shi ([5]), our bases will be of Lagrange instead of Hermite type, by which we extend the range of stability from s ∈ (2, 5 2 ) to s ∈ (1, 5 2 ). Since the latter range includes s = 2, with respect to the present basis, the stiffness matrices of fourth order elliptic problems are uniformly well-conditioned.2000 Mathematics Subject Classification. 41A15, 65D07, 65T60, 65N30, 41A63.

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Local Lagrange Interpolation with Bivariate Splines of Arbitrary Smoothness

Cited by 20 publications

References 21 publications

A local Lagrange interpolation method based on $C^{1}$ cubic splines on Freudenthal partitions

A local Lagrange interpolation method based on $C^{1}$ cubic splines on Freudenthal partitions

Investigation of Smoothness-Increasing Accuracy-Conserving Filters for Improving Streamline Integration through Discontinuous Fields

Hierarchical Riesz Bases for Hs(Ω), 1 < s < 5/2

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