Constructive Approximation

, Volume 17, Issue 4, pp 561–588

The Finite Element Method on the Sierpinski Gasket

  • Michael Gibbons
  • Arjun Raj
  • Robert S. Strichartz
Article

DOI: 10.1007/s00365-001-0010-z

Cite this article as:
Gibbons, M., Raj, A. & Strichartz, R. Constr. Approx. (2001) 17: 561. doi:10.1007/s00365-001-0010-z
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Abstract.

For certain classes of fractal differential equations on the Sierpinski gasket, built using the Kigami Laplacian, we describe how to approximate solutions using the finite element method based on piecewise harmonic or piecewise biharmonic splines. We give theoretical error estimates, and compare these with experimental data obtained using a computer implementation of the method (available at the web site http://mathlab.cit.cornell.edu/\sim gibbons). We also explain some interesting structure concerning the spectrum of the Laplacian that became apparent from the experimental data.

Key words. Finite element method, Sierpinski gasket, Fractal differential equations. AMS Classification. Primary: 28A80, 65N30.

Copyright information

© Springer-Verlag New York Inc. 2001

Authors and Affiliations

  • Michael Gibbons
    • 1
  • Arjun Raj
    • 3
  • Robert S. Strichartz
    • 5
  1. 1.Mathematics Department Manhattan College Bronx, NY 10471 USA mgibbons17@aol.comUS
  2. 2. Current address: IBM North Castle Drive, 2A-70C Armonk, NY 10504 USAUS
  3. 3.Mathematics Department University of California Berkeley, CA 94720 USA arhoon@hotmail.comUS
  4. 4. Current address: (from September, 2001) Courant Institute 251 Mercer Street New York, NY 10012 USAUS
  5. 5.Mathematics Department Malott Hall Cornell University Ithaca, NY 14853 USA str@math.cornell.eduUS