Zajednički seminar FO i MO

A large number of processes are accurately described by operator functions with a nonlinear dependence of a spectral parameter, but a linear dependence on the field. Problems involving operator functions arise from important applications in fluid dynamics, acoustics, quantum mechanics, and electromagnetic field theory.

In this talk, I show examples for which the finite-element approximation of the spectrum of a differential operator may fail. These examples illustrate the importance of a mathematical analysis of the underlying PDE. I focus on Galerkin spectral approximation theory for operator functions with periodic coefficients and the computation of physically meaningful solutions. The main applications are metallic photonic crystals and metamaterials, which are promising materials for controlling and manipulating electromagnetic waves. In these materials, the spectral parameter is usually related to the time frequency and the Floquet-Bloch wave vector is a parameter. This leads to a rational spectral problem when the frequency dependence of a Lorentz material model is included. A different approach is based on a quadratic spectral problem in the amplitude of the Floquet-Bloch wave vector. The strengths and weaknesses of both approaches are discussed in the talk.

We use high-order finite element methods with curvilinear elements to discretize the quadratic and the rational eigenvalue problem. The resulting matrix problems are transformed into linear eigenvalue problems and approximate eigenpairs are computed with a Krylov space method. Two different linearization techniques for rational eigenvalue problems and the connection between the two spectral problems will be discussed. The presentation illustrates the importance of the interplay between physical modeling, spectral theory, finite element discretization, and linear algebra.

Hrvoje Buljan i Luka Grubišić