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Macro-elements and stable local bases for splines on Powell-Sabin triangulations

Authors: Ming-Jun Lai and Larry L. Schumaker
Journal: Math. Comp. 72 (2003), 335-354
MSC (2000): Primary 41A15, 65M60, 65N30
Published electronically: July 22, 2001
MathSciNet review: 1933824
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Macro-elements of arbitrary smoothness are constructed on Powell-Sabin triangle splits. These elements are useful for solving boundary-value problems and for interpolation of Hermite data. It is shown that they are optimal with respect to spline degree, and we believe they are also optimal with respect to the number of degrees of freedom. The construction provides local bases for certain superspline spaces defined over Powell-Sabin refinements. These bases are shown to be stable as a function of the smallest angle in the triangulation, which in turn implies that the associated spline spaces have optimal order approximation power.

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Additional Information

Ming-Jun Lai
Affiliation: Department of Mathematics, The University of Georgia, Athens, Georgia 30602

Larry L. Schumaker
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240

Keywords: Macro-elements, stable bases, spline spaces
Received by editor(s): March 8, 2000
Received by editor(s) in revised form: January 31, 2001
Published electronically: July 22, 2001
Additional Notes: The first author was supported by the National Science Foundation under grant DMS-9870187
The second author was supported by the National Science Foundation under grant DMS-9803340 and by the Army Research Office under grant DAAD-19-99-1-0160
Article copyright: © Copyright 2001 American Mathematical Society