Linear and Nonlinear Eigenproblems for PDEs

Abstract

The workshop discussed the numerical solution of linear and nonlinear eigenvalue problems for partial differential equations. It included the theoretical analysis the development of new (adaptive) methods, the iterative solution of the algebraic problems as well as the application in many areas of science and engineering. Mathematics Subject Classification (2000): 65N25, 65F15, 15A18. Introduction by the Organisers The workshop Linear and Nonlinear Eigenproblems for PDEs, organised by Andrew Knyazev (University of Colorado, Denver), Volker Mehrmann (Technical University, Berlin), John E. Osborn (University of Maryland, College Park) and Jinchao Xu (Pennsylvania State University, University Park) was held August 9th – August 15th, 2009. This meeting was well attended with over 40 participants with broad geographic representation from all continents. Numerical solution of linear and nonlinear eigenvalue problems for partial differential equations is an important task in many application areas such as: • dynamics of electromagnetic fields; • electronic structure calculations; • band structure calculations in photonic crystals; • vibration analysis of heterogeneous material structures; • particle accelerator simulations; 2026 Oberwolfach Report 37/2009 • vibrations and buckling in mechanics, structural dynamics; • neutron flow simulations in nuclear reactors. It involves research in several different areas of mathematics ranging from matrix computation to modern numerical treatment of partial differential equations. Major new research developments in the area of PDE eigenvalue problems that have taken place in recent years include the following: • meshless and generalized finite element method approximation methods; • adaptive finite element methods; • methods for polynomial and other nonlinear eigenvalue problems; • a priori and a posteriori eigenvalue and eigenvector error estimation; • convergence theory for preconditioned and inexact eigensolvers; • multigrid, domain decomposition and incomplete factorization based preconditioning for eigenproblems; • public software implementing efficient eigensolvers for parallel computers. Furthermore, important progress has been made for some of the challenging problems in this area, e.g., efficient solvers have been developed for some classes of non-selfadjoint and non-linear eigenvalue problems. On the other hand, many difficult questions remain open even for linear eigenvalue problems as practically interesting engineering applications demand high accuracy of the models, which often leads to ill-conditioned problems of enormous sizes. We feel that the workshop summed up the recent developments in the different communities in this area and indicated new and fruitful directions. The purpose of this meeting was to bring together researchers with diverse background in matrix computations, numerical methods for PDEs, and relevant application areas, to present the state of the art in the area, and to exchange ideas and approaches for further development. Senior investigators as well as a significant number of young researchers participated the workshop which format was ideal to stimulate research in this increasingly important and rapidly developing area.