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MATHEMATICS OF COMPUTATION Volume 75, Number 254, Pages 711–726 S 0025-5718(05)01813-2 Article electronically published on December 30, 2005 SMOOTH MACRO-ELEMENTS ON POWELL-SABIN-12 SPLITS LARRY L. SCHUMAKER AND TATYANA SOROKINA Abstract. Macro-elements of smoothness Cr are constructed on Powell- Sabin-12 splits of a triangle for all r ≥ 0. These new elements complement those recently constructed on Powell-Sabin-6 splits and can be used to con- struct convenient superspline spaces with stable local bases and full approxi- mation power that can be applied to the solution of boundary-value problems and for interpolation of Hermite data. 1. Introduction A bivariate macro-element defined on a triangle T consists of a finite-dimensional linear space S defined on T , and a set Λ of linear functionals forming a basis for the dual of S . Usually the space S is chosen to be a space of polynomials or a space of piecewise polynomials defined on some subtriangulation of T . The members of Λ, 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 macro-element 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 ∈ S such that λs = λf for all λ ∈ Λ. We say that a macro-element has smoothness Cr provided that if the element is used to construct an interpolating function locally on each triangle of a triangulation , then the resulting piecewise function is Cr c