In this tutorial, we cover the basics of solving dynamic equations on the Sierpinski Gasket through numerical techniques. The work is divided up into three large lessons; lesson 1 covers the basics of the construction of the Sierpinski Gasket, and the construction of the Laplacian. The Laplacian can be defined on the Sierpinski Gasket through an explicit construction of a harmonic calculus, originally due to Jun Kigami; we follow his basic approach, while trying to keep the material accessible to those new in the area. Lesson 2 covers a few numerical algorithms that help solve differential equations, including an overview of a finite element method. Lesson 3 delves more in-depth into the finite element method. It illustrates how to use the finite element method to generate important function, such as the Green's Function and eigenfunctions.
As the Sierpinski Gasket is not a domain that one normally works in, special care is taken to illustrate the algorithms and methods visually. We often also point out the similarities between various concepts on the Sierpinski Gasket, and those in Euclidean spaces.
Finally, we provide custom-written software for producing all the graphs in the tutorials; this general software can be used for further exploration and solving differential equations on the Sierpinski Gasket. The software utilizes freely available tools to allow one to produce detailed 3d graphs. Lesson 3 discusses using the software.
|Author Institutional Affiliation
Department of Mathematics
University of Nebraska-Lincoln