Summary: Smooth MacroElements Based
on PowellSabin Triangle Splits
Peter Alfeld 1) and Larry L. Schumaker 2)
Abstract. Macroelements of smoothness C r on PowellSabin triangle splits
are constructed for all r – 0. These new elements are improvements on el
ements constructed in  in that certain unneeded degrees of freedom have
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 degrees of freedom, are usually taken to be point evaluations of derivatives.
A macroelement defines a local interpolation scheme. In particular, if f is
a sufficiently smooth function, then we can define the corresponding interpolant
as the unique function s 2 S such that –s = –f for all – 2 \Lambda. We say that a
macroelement has smoothness C r provided that if the element is used to construct
an interpolating function locally on each triangle of a triangulation 4, then the
resulting piecewise function is C r continuous globally.