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Splines on fractals

Published online by Cambridge University Press:  16 October 2000

ROBERT S. STRICHARTZ  and
MICHAEL USHER
ROBERT S. STRICHARTZ
Affiliation:
Mathematics Department, Malott Hall, Cornell University, Ithaca, NY 14853, U.S.A.; e-mail: [email protected]
MICHAEL USHER
Affiliation:
Mathematics Department, University of California, Berkeley, CA 94720, U.S.A.; e-mail: [email protected]
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Abstract

A general theory of piecewise multiharmonic splines is constructed for a class of fractals (post-critically finite) that includes the familiar Sierpinski gasket, based on Kigami's theory of Laplacians on these fractals. The spline spaces are the analogues of the spaces of piecewise Cj polynomials of degree 2j + 1 on an interval, with nodes at dyadic rational points. We give explicit algorithms for effectively computing multiharmonic functions (solutions of Δj+1u = 0) and for constructing bases for the spline spaces (for general fractals we need to assume that j is odd), and also for computing inner products of these functions. This enables us to give a finite element method for the approximate solution of fractal differential equations. We give the analogue of Simpson's method for numerical integration on the Sierpinski gasket. We use splines to approximate functions vanishing on the boundary by functions vanishing in a neighbourhood of the boundary.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 129 , Issue 2 , September 2000 , pp. 331 - 360
Copyright
2000 Cambridge Philosophical Society

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