Analysis of Boundary Element Methods

The workshop Analysis of Boundary Element Methods , organized by Martin Costabel (Rennes) and Ernst P. Stephan (Hannover). This meeting brought together 46 experts in numerical analysis. Boundary element methods (BEM) are well established numerical methods with a wide range of applications. There are still many challenging problems, the aim of this workshop was to coordinate the efforts to tackle these problems. A central theme of many talks was the error analysis. In the following we focus on some central aspects. 1. BEM for time dependent problems: Space-time boundary integral equations using integral representations with the fundamental solution of hyperbolic problems have become increasingly important in elastodynamics and electrodynamics. New results were presented concerning error analysis, stability and performance of numerical algorithms. Special attention was given to efficient quadrature of retarded potentials and also to the method of convolution quadrature for the computation of transient acoustic waves. 2.Stability at high frequencies: The numerical modeling of scattering problems at higher frequencies poses practical challenges due to the need to approximate highly oscillating functions. In addition, the analysis of many of the standard BEM loses usefulness in the high frequency range because the constants in the stability and error estimates either grow rapidly with the frequency of have unspecified dependency on the frequency. New boundary element methods have been presented whose stability deteriorates only slowly with frequency or are even unconditionally stable for all frequencies. 3. Fast algorithms: Multilevel approaches have produced several families of highly efficient algorithms. Several talks reported on the applications of the fast multipole method and on iterative techniques based on multigrid methods, H-matrices or on Schwarz preconditioners. 4. New concepts for BEM: Topics have been here: sparse p-version BEM with random loading, meshless methods for pseudodifferential equations, adaptive procedures, least squares FEM-BEM coupling, p-version applied to EFIE, BEM for eigenvalue problems. 5. Specific applications: Here the following have been addressed: Source identification and source reconstructions in acoustics with the help of inverse boundary elements, BEM for Maxwell's equations (currents and charges boundary integral equations, shape optimization of integrated lens antennas, electromagnetic scattering by a dielectric body), BEM for option pricing. The list of participants included several of the founders of the discipline of mathematical analysis of BEM, but also some PhD students. This mix of both established and younger researchers created a particularly stimulating research atmosphere.