20190412
Everybody is familiar with the concept of eigenvalues of an n times n matrix. In this talk, we consider the nonlinear eigenvalue problem. These are problems for which the eigenvalue parameter appears in a nonlinear way in the equation. The last decade, the number of applications is increasing. In physics, the Schroedinger equation for determining the bound states in a semiconductor device, introduces terms with square roots of different shifts of the eigenvalue. In mechanical and civil engineering, new materials often have nonlinear damping properties. For the vibration analysis of such materials, this leads to nasty functions of eigenvalue in the system matrix.
One particular example is the sandwhich beam problem, where a layer of damping material is sandwhiched between two layers of steel. Another example is the stability analysis of the noise produced by burners in a combustion chamber. The burners lead to a boundary condition with delay terms (exponentials of the eigenvalue).
We often receive the question: “How can we solve a nonlinear eigenvalue problem?” This talk explains the different steps to be taken for using Krylov methods. The general approach works as follows: 1) approximate the nonlinearity by a rational function; 2) rewrite this rational eigenvalue problem as a linear eigenvalue problem and then 3) solve this by a Krylov method. We explain each of the three steps in detail.
For step one, we explain the tools from approximation theory that can be used: spectral approximation, potential theory, Adaptive AntoulasAnderson, Padé approximation. For steps 2 and 3, we explain which bases of rational polynomials should be used for efficient application of Krylov methods. Numerical examples illustrate the different choices.
Technische Universität Darmstadt
Graduate School CE
Dolivostraße 15
D64293 Darmstadt

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