Helmholtz Equation

The group of the physicist Prof. Dr. A. Richter uses experiments with microwaves to study quantum manifestation of classical chaos. To this end, they build super-conductive microwave resonators and measure the spectrum of eigenvalues. For details click here.

As an example the following picture shows a flat resonator for a Pascalian snail:

The physical experiment is mathematically described by the stationary Helmholtz equation:

The finite element discretisation of this equation leads to a generalised eigenvalue problem:

Using Web-Splines of degree 5 on a 75x75 grid we are able to compute 2000 eigenvalues with relative error less then 10^-4 in half an hour on a 2 GHz standard PC .

Here are some examples of the corresponding eigenfunctions:

first eigenfunction, k² = 106.6774

second eigenfunction, k2 = 254.2339

third eigenfunction, k2 = 286.0975

tenth eigenfunction, k² = 960.0726

fiftieth eigenfunction, k² = 4119.0409

Literature

See Publications for a complete list of WEB-publications.

Author: Bernhard Mößner ; Last modification: 2003/10/17 16:17:04 UTC.