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The WEB-Method

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:

Microwave resonator shaped as Pascalian snail

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

Helmholz equation

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

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
1. eigenfunction

second eigenfunction, k2 = 254.2339
2. eigenfunction
third eigenfunction, k2 = 286.0975
3. eigenfunction

tenth eigenfunction, k2 = 960.0726
10. eigenfunction

fiftieth eigenfunction, k2 = 4119.0409
50. eigenfunction


A. Richter: Playing Billiards with Microwaves - Quantum Manifestations of Classical Chaos, The IMA Volumes in Mathematics and its Application, Vol. 109, Springer 1999.

See Publications for a complete list of WEB-publications.

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

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