The exponential accuracy of Fourier and Chebyshev differencing methods
SIAM Journal on Numerical Analysis
A pseudospectral approach for polar and spherical geometries
SIAM Journal on Scientific Computing
Deflation Techniques for an Implicitly Restarted Arnoldi Iteration
SIAM Journal on Matrix Analysis and Applications
Eigenmodes of Isospectral Drums
SIAM Review
Matrix computations (3rd ed.)
Spectral methods in MatLab
Computation of conformal maps by modified schwarz-christoffel transformations
Computation of conformal maps by modified schwarz-christoffel transformations
A Multipole Method for Schwarz--Christoffel Mapping of Polygons with Thousands of Sides
SIAM Journal on Scientific Computing
The Accuracy of the Chebyshev Differencing Method for Analytic Functions
SIAM Journal on Numerical Analysis
Reviving the Method of Particular Solutions
SIAM Review
Computing eigenfunctions on the Koch Snowflake: a new grid and symmetry
Journal of Computational and Applied Mathematics
Practical Extrapolation Methods: Theory and Applications
Practical Extrapolation Methods: Theory and Applications
Computing eigenmodes ofelliptic operators using radial basis functions
Computers & Mathematics with Applications
A Symmetry-Based Decomposition Approach to Eigenvalue Problems
Journal of Scientific Computing
| Hi-index | 7.29 |
A method for the computation of eigenfrequencies and eigenmodes of fractal drums is presented. The approach involves first conformally mapping the unit disk to a polygon approximating the fractal and then solving a weighted eigenvalue problem on the unit disk by a spectral collocation method. The numerical computation of the complicated conformal mapping was made feasible by the use of the fast multipole method as described in [L. Banjai, L.N. Trefethen, A multipole method for Schwarz-Christoffel mapping of polygons with thousands of sides, SIAM J. Sci. Comput. 25(3) (2003) 1042-1065]. The linear system arising from the spectral discretization is large and dense. To circumvent this problem we devise a fast method for the inversion of such a system. Consequently, the eigenvalue problem is solved iteratively. We obtain eight digits for the first eigenvalue of the Koch snowflake and at least five digits for eigenvalues up to the 20th. Numerical results for two more fractals are shown.