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

Abstract

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.

arXiv comment31 pages, 2 figures. Fixed a typo. Article published in M3AS

References in corpus (3)

• Robust Multigrid for Isogeometric Analysis Based on Stable Splittings of Spline Spaces
• Approximation error estimates and inverse inequalities for B-splines of maximum smoothness
• Robust multigrid solvers for the biharmonic problem in isogeometric analysis

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• On best constants in $L^2$ approximation
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• On the normalization of trigonometric and hyperbolic B-splines
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