Inf-sup stability of isogeometric Taylor-Hood and Sub-Grid methods for the Stokes problem with hierarchical splines

In this article, we prove the inf-sup stability of an adaptive isogeometric discretization of the Stokes problem. The discretization is based on the hierarchical generalization of the isogeometric Taylor-Hood and Sub-Grid elements, which were described by Bressan & Sangalli (2013, Isogeometric discretizations of the Stokes problem: stability analysis by the macroelement technique. IMA J. Numer. Anal., 33, 629- 651) for tensor-product splines. In order to extend the existing proof to the hierarchical setting, we need to adapt some of the steps considerably. In particular, the required local approximation estimate is obtained by analysing the properties of the quasi-interpolant of Speleers & Manni (2016, Effortless quasi-interpolation in hierarchical spaces. Numer. Math., 132, 155-184) with respect to certain Sobolev norms. In addition to the theoretical results, we also perform numerical tests in order to analyse the dependency of the inf-sup constant on the mesh regularity assumptions. Finally, the article also presents a numerical convergence test of the resulting adaptive method on a T-shaped domain

Bivariate hierarchical Hermite spline quasi--interpolation

Spline quasi-interpolation (QI) is a general and powerful approach for the construction of low cost and accurate approximations of a given function. In order to provide an efficient adaptive approximation scheme in the bivariate setting, we consider quasi-interpolation in hierarchical spline spaces. In particular, we study and experiment the features of the hierarchical extension of the tensor-product formulation of the Hermite BS quasi-interpolation scheme. The convergence properties of this hierarchical operator, suitably defined in terms of truncated hierarchical B-spline bases, are analyzed. A selection of numerical examples is presented to compare the performances of the hierarchical and tensor-product versions of the scheme

Characterization of bivariate hierarchical quartic box splines on a three-directional grid

International audienceWe consider the adaptive refinement of bivariate quartic C 2-smooth box spline spaces on the three-directional (type-I) grid G. The polynomial segments of these box splines belong to a certain subspace of the space of quar-tic polynomials, which will be called the space of special quartics. Given a bounded domain Ω ⊂ R 2 and finite sequence (G ℓ) ℓ=0,...,N of dyadically refined grids, we obtain a hierarchical grid by selecting mutually disjoint cells from all levels such that their union covers the entire domain. Using a suitable selection procedure allows to define a basis spanning the hierarchical box spline space. The paper derives a characterization of this space. Under certain mild assumptions on the hierarchical grid, the hierarchical spline space is shown to contain all C 2-smooth functions whose restrictions to the cells of the hierarchical grid are special quartic polynomials. Thus, in this case we can give an affirmative answer to the completeness questions for the hierarchical box spline basis

Weighted Quasi Interpolant Spline Approximations: Properties and Applications

Continuous representations are fundamental for modeling sampled data and performing computations and numerical simulations directly on the model or its elements. To effectively and efficiently address the approximation of point clouds we propose the Weighted Quasi Interpolant Spline Approximation method (wQISA). We provide global and local bounds of the method and discuss how it still preserves the shape properties of the classical quasi-interpolation scheme. This approach is particularly useful when the data noise can be represented as a probabilistic distribution: from the point of view of nonparametric regression, the wQISA estimator is robust to random perturbations, such as noise and outliers. Finally, we show the effectiveness of the method with several numerical simulations on real data, including curve fitting on images, surface approximation and simulation of rainfall precipitations

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