Computing eigenvalues with higher order Finite Elements |
|
The workgroup of Prof. Dr. A. Richter (Institut für Kernphysik, TU Darmstadt) uses experiments with
microwaves to investigate quantum manifestations of classical chaos.
Therefore superconducting microwave-resonatores are manufactured such that the spectrum of
eigenfrequences can be determind by measuremnet.
A
more precisely descriptionen of these experiments
can be found here.
|
|
The following picture shows an example of a flat resonator for a Pascalian snail:
|
|
|
The experiment can mathematically be modeld by a stationary Helmholtz-equation:
|
|
|
Discretizing with a Finite Element method yields a generalized eigenvalue problem:
|
|
|
Here Web-Splines are used as Finite Elements.
Using basis functions of degree 5 on a 75x75 grid
2000 eigenvalues with a relative error less then
10-4
can be computed in one hour on a 2GHz standard PC.
|
|
The following pictures show some examples of corresponding eigenfunctions:
|
|
|
Literatur
|
|
A. Richter: Playing Billiards with Microwaves - Quantum Manifestations of Classical Chaos,
The IMA Volumes in Mathematics and its Application, Vol. 109, Springer 1999.
|
