Sharp error estimates for spline approximation: explicit constants, nn-widths, and eigenfunction convergence

In this paper we provide a priori error estimates in standard Sobolev (semi-)norms for approximation in spline spaces of maximal smoothness on arbitrary grids. The error estimates are expressed in terms of a power of the maximal grid spacing, an appropriate derivative of the function to be approximated, and an explicit constant which is, in many cases, sharp. Some of these error estimates also hold in proper spline subspaces, which additionally enjoy inverse inequalities. Furthermore, we address spline approximation of eigenfunctions of a large class of differential operators, with a particular focus on the special case of periodic splines. The results of this paper can be used to theoretically explain the benefits of spline approximation under $k$-refinement by isogeometric discretization methods. They also form a theoretical foundation for the outperformance of smooth spline discretizations of eigenvalue problems that has been numerically observed in the literature, and for optimality of geometric multigrid solvers in the isogeometric analysis context.Comment: 31 pages, 2 figures. Fixed a typo. Article published in M3A

Quadratic Spline Quasi-interpolants on Powell-Sabin Partitions

2004-16International audienceIn this paper we address the problem of constructing quasi-interpolants in the space of quadratic Powell-Sabin splines on nonuniform triangulations. Quasi-interpolants of optimal approximation order are proposed and numerical tests are presented

On a class of polynomial triangular macro-elements

AbstractIn this paper we present a new class of polynomial triangular macro-elements of arbitrary degree which are an extension of the classical Clough-Tocher cubic scheme. Their most important property is that the degree plays the role of a tension parameter, since these macro elements tend to the plane interpolating the vertices data. Graphical examples showing their use in scattered data interpolation are reported

Tchebycheffian B-splines in isogeometric Galerkin methods

Tchebycheffian splines are smooth piecewise functions whose pieces are drawn from (possibly different) Tchebycheff spaces, a natural generalization of algebraic polynomial spaces. They enjoy most of the properties known in the polynomial spline case. In particular, under suitable assumptions, Tchebycheffian splines admit a representation in terms of basis functions, called Tchebycheffian B-splines (TB-splines), completely analogous to polynomial B-splines. A particularly interesting subclass consists of Tchebycheffian splines with pieces belonging to null-spaces of constant-coefficient linear differential operators. They grant the freedom of combining polynomials with exponential and trigonometric functions with any number of individual shape parameters. Moreover, they have been recently equipped with efficient evaluation and manipulation procedures. In this paper, we consider the use of TB-splines with pieces belonging to null-spaces of constant-coefficient linear differential operators as an attractive substitute for standard polynomial B-splines and rational NURBS in isogeometric Galerkin methods. We discuss how to exploit the large flexibility of the geometrical and analytical features of the underlying Tchebycheff spaces according to problem-driven selection strategies. TB-splines offer a wide and robust environment for the isogeometric paradigm beyond the limits of the rational NURBS model.Comment: 35 pages, 18 figure

A tension approach to controlling the shape of cubic spline surfaces on FVS triangulations

We propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to C1 cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods

Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis

In this paper we provide a priori error estimates with explicit constants for both the $L^2$-projection and the Ritz projection onto spline spaces of arbitrary smoothness defined on arbitrary grids. This extends the results recently obtained for spline spaces of maximal smoothness. The presented error estimates are in agreement with the numerical evidence found in the literature that smoother spline spaces exhibit a better approximation behavior per degree of freedom, even for low smoothness of the functions to be approximated. First we introduce results for univariate spline spaces, and then we address multivariate tensor-product spline spaces and isogeometric spline spaces generated by means of a mapped geometry, both in the single-patch and in the multi-patch case.Comment: 39 pages, 4 figures. Improved the presentation. Article published in Numerische Mathemati

Adaptive refinement with locally linearly independent LR B-splines: Theory and applications

In this paper we describe an adaptive refinement strategy for LR B-splines. The presented strategy ensures, at each iteration, local linear independence of the obtained set of LR B-splines. This property is then exploited in two applications: the construction of efficient quasi-interpolation schemes and the numerical solution of elliptic problems using the isogeometric Galerkin method.Comment: 23 pages, 14 figure

Convergence of univariate non-stationary subdivision schemes via asymptotical similarity

A new equivalence notion between non-stationary subdivision schemes, termed asymptotical similarity, which is weaker than asymptotical equivalence, is introduced and studied. It is known that asymptotical equivalence between a non-stationary subdivision scheme and a convergent stationary scheme guarantees the convergence of the non-stationary scheme. We show that for non-stationary schemes reproducing constants, the condition of asymptotical equivalence can be relaxed to asymptotical similarity. This result applies to a wide class of non-stationary schemes of importance in theory and applications

Best low-rank approximations and Kolmogorov n-widths

We relate the problem of best low-rank approximation in the spectral norm for a matrix $A$ to Kolmogorov $n$-widths and corresponding optimal spaces. We characterize all the optimal spaces for the image of the Euclidean unit ball under $A$ and we show that any orthonormal basis in an $n$-dimensional optimal space generates a best rank-$n$ approximation to $A$. We also present a simple and explicit construction to obtain a sequence of optimal $n$-dimensional spaces once an initial optimal space is known. This results in a variety of solutions to the best low-rank approximation problem and provides alternatives to the truncated singular value decomposition. This variety can be exploited to obtain best low-rank approximations with problem-oriented properties.Comment: 25 pages, 1 figur