Stable splitting of bivariate spline spaces by Bernstein-Bézier methods

We develop stable splitting of the minimal determining sets for the spaces of bivariate C1 splines on triangulations, including a modified Argyris space, Clough-Tocher, Powell-Sabin and quadrilateral macro-element spaces. This leads to the stable splitting of the corresponding bases as required in Böhmer's method for solving fully nonlinear elliptic PDEs on polygonal domains

Gaussian quadrature for C1C^1 cubic Clough-Tocher macro-triangles

A numerical integration rule for multivariate cubic polynomials over n-dimensional simplices was designed by Hammer and Stroud [14]. The quadrature rule requires n + 2 quadrature points: the barycentre of the simplex and n + 1 points that lie on the connecting lines between the barycentre and the vertices of the simplex. In the planar case, this particular rule belongs to a two-parameter family of quadrature rules that admit exact integration of bivariate polynomials of total degree three over triangles. We prove that this rule is exact for a larger space, namely the C1 cubic Clough-Tocher spline space over macro-triangles if and only if the split-point is the barycentre. This results into a factor of three reduction in the number of quadrature points needed to integrate the Clough-Tocher spline space exactly

Unterteilungsmethoden für Triangulierungen und bivariate Splineinterpolation

In dieser Arbeit beschreiben wir eine effiziente Unterteilungsmethode für die Interpolation mit bivariaten Splines auf beliebigen Triangulierungen. Bei unserer Methode müssen nur wenige Dreiecke einer Triangulierung Δ unterteilt werden. [...] weiter s. Originaldokumen

Scattered data fitting on surfaces using projected Powell-Sabin splines

We present C1 methods for either interpolating data or for fitting scattered data associated with a smooth function on a two-dimensional smooth manifold Ω embedded into R3. The methods are based on a local bivariate Powell-Sabin interpolation scheme, and make use of local projections on the tangent planes. The data fitting method is a two-stage method. We illustrate the performance of the algorithms with some numerical examples, which, in particular, confirm the O(h3) order of convergence as the data becomes dens

Generalized Finite Element Systems for smooth differential forms and Stokes problem

We provide both a general framework for discretizing de Rham sequences of differential forms of high regularity, and some examples of finite element spaces that fit in the framework. The general framework is an extension of the previously introduced notion of Finite Element Systems, and the examples include conforming mixed finite elements for Stokes' equation. In dimension 2 we detail four low order finite element complexes and one infinite family of highorder finite element complexes. In dimension 3 we define one low order complex, which may be branched into Whitney forms at a chosen index. Stokes pairs with continuous or discontinuous pressure are provided in arbitrary dimension. The finite element spaces all consist of composite polynomials. The framework guarantees some nice properties of the spaces, in particular the existence of commuting interpolators. It also shows that some of the examples are minimal spaces.Comment: v1: 27 pages. v2: 34 pages. Numerous details added. v3: 44 pages. 8 figures and several comments adde

Interpolation by spline spaces on classes of triangulations

We describe a general method for constructing triangulations Δ which are suitable for interpolation by Srq(Δ),

Interpolation and scattered data fitting on manifolds using projected Powell–Sabin splines

We present methods for either interpolating data or for fitting scattered data on a two-dimensional smooth manifold. The methods are based on a local bivariate Powell-Sabin interpolation scheme, and make use of a family of charts {(Uξ , ξ)}ξ∈ satisfying certain conditions of smooth dependence on ξ. If is a C2-manifold embedded into R3, then projections into tangent planes can be employed. The data fitting method is a two-stage method. We prove that the resulting function on the manifold is continuously differentiable, and establish error bounds for both methods for the case when the data are generated by a smooth function

A C r Trivariate Macro-Element Based on the Alfeld Split of Tetrahedra

Abstract We construct trivariate macro-elements of class C r for any r ≥ 1 over the Alfeld refinement of any tetrahedral partition in R 3 . In our construction, the degree of polynomials used for these macro-elements is the lowest possible. We also give the dimension formula for the subspace of consisting of these macro-elements