Parameter-dependent Parallel Block Sparse Arnoldi and Döhler Algorithms on Distributed Systems

We summarize the basics and first results of the analyses within our ZIB Bridge Project and give an outlook on further studies broadening the usage of hardware acceleration within the Finite Element Method (FEM) based solution of Maxwell’s equations. 1 Solving the Generalized Eigenvalue Problem The main focus of our project is on the solution of the eigenvalue problem originating from Maxwell’s equations in nano-optics. The basic example of such problems are so-called resonance problems [18]. These yield generalized eigenvalue problems, of which the formulation and different solution approaches are discussed in the following. We start by stating the problem and introducing a typical example, namely nanoholes in photonic crystals [1]. Furthermore, we describe three different algorithms to solve the generalized eigenvalue problem: the standard Arnoldi iteration, the recently introduced [16] and extended [8] FEAST algorithm and a Döhler algorithm [2]. 1.1 Problem Formulation In the FEM formulation [14] for Maxwell’s equations [4], we face the problem of solving the generalized eigenvalue problem: