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A characterization of supersmoothness of multivariate splines. (English) Zbl 1447.41014

Piecewise polynomials in one dimension (the classical polynomial splines) and in higher dimensions (two dimensions in particular, used as finite elements and for CAGD for instance) are fundamental tools in approximation theory. They are used for modelling solutions of PDEs, they are used for quasi-interpolation of data and, to give a further example, they are used for creating surfaces in design. The definition of the splines rests on the partitioning of the space into cells (typically triangles), where we just have polynomials, and the joining of these polynomials to identify a function over the complete domain or space.
These splines are usually piecewise polynomials of a certain, given differentiability at the edges and vertices (and of course smooth in the inside of the triangles). Both polynomial degree and this differentiability characterise the spline. Under certain conditions we have a higher order of smoothness than expected at the points where the polynomials are joined, and this is called super smoothness.
The purpose of this paper is to characterise this phenomenon in a setting which is as general as possible. This is carried out by considering general, smooth pieces of a function (usually polynomials) and the local behaviour of the approximant at the vertices. The authors use a characterisation of the degeneracy of a polynomial space over the cell (or finite element) in order to find the degree of supersmoothness at the vertices. Degeneracy means in this context that the approximant happens not to be a piecewise polynomial (of different polynomial pieces) but actually a straight polynomial altogether.
Maximal orders of supersmoothness are derived and the results are applied to many famous forms of splines or elements (Alfeld, Clough-Tocher, Worsey-Farin).

MSC:

41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
41A15 Spline approximation
65D07 Numerical computation using splines
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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References:

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