Adaptive methods for nonlinear eigenvalue problems with parameters
Final Report Abstract
Let D ⊂ R denotes an open interval and T (λ) : D → H be a selfadjoint and bounded operator on a real Hilbert space H. In the following we consider the infinite dimensional nonlinear PDE eigenvalue problems (NLEVPs): Find u ∈ H \{0} and λ ∈ D such that T (λ)u = 0, with λ ∈ D being an eigenvalue of T and u the corresponding eigenfunction. Special cases of NLEVPS include elliptic linear and nonoverdamped quadratic eigenvalue problems. Problems of this type arise in many applications, e.g., in structural dynamics while analyzing the behavior of structures with physical damping, fluid-structure interaction problemor acoustic problems with absorbing boundary conditions or in electronic structure calculation for quantum dots . Sometimes the underlying nonlinear eigenvalue problems are dependent not only on the spectral parameter, but additionally on a large range of physical parameters e.g., frequency in the case of viscoelastic materials or wave vectors in the band structure calculations for photonic crystals. The abundance of nonlinear eigenvalue problems arise from the discretization of differential and integral equation. This work characterized the asymptotic behavior of the errors and gave asymptotic rates of convergence for the eigenfunctions corresponding to the smallest eigenvalue of NLEVPs through a priori error analysis. Computable, effcient and reliable bounds on the actual eigenvalue/eigenfunction errors through a posteriori error analysis are still under investigations.
Publications
- A Story on Adaptive Finite Element Computations for Elliptic Eigenvalue Problems, in Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory, Festschrift in Honor of Volker Mehrmann, eds. P. Benner et al., Springer International Publishing Switzerland, pp. 223-255, 2015
A. Międlar
- An algorithm for computing minimal Gersgorin sets, Numer. Linear Algebra Appl. 23(2):272-290, 2016
V. Kostić, A. Międlar and Lj. Cvetković
(See online at https://doi.org/10.1002/nla.2024)
