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cta(.tex) (as of Nov. 13, 2000) T E X'ed at 13:32 on 15 November 2000 Smooth MacroElements Based
 

Summary: cta(.tex) (as of Nov. 13, 2000) T E X'ed at 13:32 on 15 November 2000
Smooth Macro­Elements Based
on Clough­Tocher Triangle Splits
Peter Alfeld 1) and Larry L. Schumaker 2)
Abstract. Macro­elements of smoothness C r on Clough­Tocher triangle splits
are constructed for all r – 0. These new elements are improvements on elements
constructed in [11] in that (disproving a conjecture made there) certain unneeded
degrees of freedom have been removed. Numerical experiments on Hermite
interpolation with the new elements are included.
x1. Introduction
A bivariate macro­element defined on a triangle T consists of a finite dimensional
linear space S defined on T and a set \Lambda of linear functionals forming a basis for the
dual of S.
It is common to choose the space S to be a space of polynomials or a space
of piecewise polynomials defined on some subtriangulation of T . The members
of \Lambda, called the degrees of freedom, are usually taken to be point evaluations of
derivatives, although here we will also work with sets of linear functionals which
pick off certain spline coefficients.
A macro­element defines a local interpolation scheme. In particular, if f is
a sufficiently smooth function, then we can define the corresponding interpolant

  

Source: Alfeld, Peter - Department of Mathematics, University of Utah

 

Collections: Mathematics