Dimensions of spline spaces over T-meshes

A T-mesh is basically a rectangular grid that allows T-junctions. In this paper, we propose a method based on Bézier nets to calculate the dimension of a spline function space over a T-mesh. When the order of the smoothness is less than half of the degree of the spline functions, a dimension formula is derived which involves only the topological quantities of the T-mesh. The construction of basis functions is briefly discussed. Furthermore, the dimension formulae for T-meshes after mesh operations, such as edge insertion and mesh merging, are also obtained.     

On the dimension of trivariate spline spaces with the highest order smoothness on 3D T-meshes

T-meshes are a type of rectangular partitions of planar domains which allow hanging vertices. Because of the special structure of T-meshes, adaptive local refinement is possible for splines defined on this type of meshes, which provides a solution for the defect of NURBS. In this paper, we generalize the definitions to the three-dimensional (3D) case and discuss a fundamental problem – the dimension of trivariate spline spaces on 3D T-meshes. We focus on a special case where splines are C ^d?1 continuous for degree d. The smoothing cofactor method for trivariate splines is explored for this situation. We obtain a general dimension formula and present lower and upper bounds for the dimension. At last, we introduce a type of 3D T-meshes, where we can give an explicit dimension formula.     

On the dimension of bivariate spline spaces over rectilinear partitions

We consider spaces of piecewise polynomials of degree n and smoothness k over a rectilinear partition of a simply connected domain of \(\mathbb{R}^2 \) . In some cases, bounds for the dimension value of the space given in the literature are improved. In addition, we provide the exact value and an explicit base of the space, if n≤k+(k+1)/D, with D+1 the maximum number of edges with different slopes emanating from a vertex of the partition.     

Alternating direction implicit orthogonal spline collocation on some non-rectangular regions with inconsistent partitions

The alternating direction implicit (ADI) method is a highly efficient technique for solving multi-dimensional dependent initial-boundary value problems on rectangles. Earlier we have used the ADI technique in conjunction with orthogonal spline collocation (OSC) for discretization in space to solve parabolic problems on rectangles and rectangular polygons. Recently, we extended applications of ADI OSC schemes to the solution of parabolic problems on some non-rectangular regions that allow for consistent nonuniform partitions. However, for many regions, it is impossible to construct such partitions. Therefore, in this paper, we show how to extend our approach further to solve parabolic problems on some non-rectangular regions using inconsistent uniform partitions. Numerical results are presented using piecewise Hermite cubic polynomials for spatial discretizations and our ADI OSC scheme for parabolic problems to demonstrate its performance on several regions.     

Adaptive finite element methods for elliptic equations over hierarchical T-meshes

Isogeometric analysis using NURBS (Non-Uniform Rational B-Splines) as basis functions gives accurate representation of the geometry and the solution but it is not well suited for local refinement. In this paper, we use the polynomial splines over hierarchical T-meshes (PHT-splines) to construct basis functions which not only share the nice smoothness properties as the B-splines, but also allow us to effectively refine meshes locally. We develop a residual-based a posteriori error estimate for the finite element discretization of elliptic equations using PHT-splines basis functions and study their approximation properties. In addition, we conduct numerical experiments to verify the theory and to demonstrate the effectiveness of the error estimate and the high order approximations provided by the numerical solution.     

Superapproximation for projections on spline spaces

Summary. In this paper general conditions are given for the superapproximation of projections on non-uniform mesh multiple knot splines in L_p-spaces. Various known results are contained as special cases.Mathematics Subject Classification (2000): 41A15, 41A25, 41A28, 65R26AcknowledgmentThe comments of the referees, among them attracting the authors attention to Theorem 2.7 in [7], are gratefully acknowledged.     

Adaptive finite element methods for elliptic equations over hierarchical T-meshes

Isogeometric analysis using NURBS (Non-Uniform Rational B-Splines) as basis functions gives accurate representation of the geometry and the solution but it is not well suited for local refinement. In this paper, we use the polynomial splines over hierarchical T-meshes (PHT-splines) to construct basis functions which not only share the nice smoothness properties as the B-splines, but also allow us to effectively refine meshes locally. We develop a residual-based a posteriori error estimate for the finite element discretization of elliptic equations using PHT-splines basis functions and study their approximation properties. In addition, we conduct numerical experiments to verify the theory and to demonstrate the effectiveness of the error estimate and the high order approximations provided by the numerical solution.     

New spline spaces with generalized tension properties

The paper describes a new space of variable degree polynomials. This space is isomorphic to ℙ₆, possesses a Bernstein like basis and has generalized tension properties in the sense that, for limit values of the degrees, its functions approximate quadratic polynomials. The corresponding space of C ³, variable degree splines is also studied. This spline space can be profitably used in the construction of shape preserving curves or surfaces. AMS subject classification (2000)  65D07, 65D17, 65D10     

Bivariate quartic spline spaces and quasi-interpolation operators

In this paper, we study two bivariate quartic spline spaces and , and present two classes of quasi-interpolation operators in the two spaces, respectively. Some results on the operators are given.     

Weak descartes systems in generalized spline spaces

A class of generalized spline spaces is introduced for which a basis of functions with local support is constructed by using a recursion relation. It is shown that this basis forms a weak Descartes system. Moreover, an interpolation property is given.     

Algorithms of wavelet compression of linear spline spaces

Splines and wavelets have been finding increasing use in the theory of information. Wavelet decompositions are used in designing efficient algorithms for processing (compression) of large information flows. If one succeeds in establishing the embeddability of spaces of splines on a sequence of sparsing/refining grids, in representing the chain of embedded spaces as a direct sum of wavelet spaces, and in realizing the base functions with the minimum length of their support, then this suggests a wavelet decomposition of the information flow, leading, in turn, to substantial savings in the computational cost. This being so, it proves possible to resolve the initial information flow into components to single out the principal and refining information flows, depending on the needs. For uniform grids on the real line, wavelet decompositions are well known. In this case, there applies the powerful technique of harmonic analysis, as well as the lifting scheme or the wavelet scheme. However, many applications require considering bounded intervals and nonuniform grids. For example, for efficient compression of nonuniform flows of information (featuring singularities or rapidly fluctuating characteristics), it is expedient to employ an adaptive nonuniform grid, which takes account of the singularities of the flow being processed. This renders possible to improve approximation of functions without complicating the computations. The previously obtained results pertained to splines on infinite grids. Making both the grid and the corresponding numerical flow infinite renders theoretical studies simpler; however, in practice, one has to deal with finite flows. This paper continues the studies initiated for finite-dimensional spaces. The purpose of this work is to built a wavelet decomposition (compression) on a nonuniform grid and develop the corresponding decomposition and reconstruction algorithms for infinite flows (with a grid on an open interval) and finite flows (with a grid on a segment) for linear spaces of splines of Lagrange type.     

Bounds on the dimensions of trivariate spline spaces

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.      

THE DIMENSION OF A CLASS OF BIVARIATE SPLINE SPACES

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Locally linearly independent bases for bivariate polynomial spline spaces

Locally linearly independent bases are constructed for the spaces S ^r _d () of polynomial splines of degree d3r+2 and smoothness r defined on triangulations, as well as for their superspline subspaces.     

Orlicz spaces,spline systems,and brownian motion

There are three results proved in this paper. The first one characterizes the Hölder classes in Orlicz spaces by the coefficients of the orthogonal spline expansions of the Franklin type. The second one gives a sharp estimate for the correlation of two random variables obtained as a composition of two Borel functions with the components of a given two-dimensional Gaussian vector. The third one is obtained with the help of the first two and it states that the Wiener measure is concentrated on the Banach space of Hölder functions with exponent 1/2 but in the norm of the Orlicz spaceL _M ^* withM(t)=expt(t ²)?1.