Koch snowflake [Lapidus, Neuberger, Renka and Griffith (1996)]
Mesh (triangle) formulation
The boundary points are indexed in the counter-clockwise order, while the interior points are indexed by the spiral order.
Finite difference scheme using 6-point for the Laplace operator:
\[ \sum_{j=2}^{7} \mathbf{u}_j = 6 \mathbf{u}_1 + \frac{3}{2}h^2 \Delta u(x_1,y_1) + O(h^4) \]
First 8 rows and columns of the Laplace matrix (finite-differnece)
| A | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| 1 | 6 | -1 | -1 | -1 | -1 | -1 | -1 | |
| 2 | -1 | 6 | -1 | -1 | -1 | |||
| 3 | -1 | -1 | 6 | -1 | ||||
| 4 | -1 | -1 | 6 | -1 | ||||
| 5 | -1 | -1 | 6 | -1 | ||||
| 6 | -1 | -1 | 6 | -1 | ||||
| 7 | -1 | -1 | -1 | 6 | ||||
| 8 | -1 | 6 |
Eigenvalue problem becomes \[ A\mathbf{u} = \lambda\mathbf{u}, \] so the eigenvector of the corresponding eigenvalue gives the eigenfunction (on the triangle mesh).