Many tasks of electromagnetic field theory are naturally posed as eigenproblems of Maxwell's equations. Typical examples are resonators and optical waveguides. The numerical solution of such problems is usually accompanied with three difficulties:

• The domain, on which the operators are defined, is unbounded.
• The Maxwell operators become nonselfadjoint, if loss or gain is considered.
• The physical solutions may be noncontinuous.

The last two problems are the topics of this project. We aim to generalize the nonlinear multigrid technique, developed in a preceding project, to the eigenproblem of Maxwell's equations. Concerning suitable variational formulations, we follow two different approaches: the first is to pose the problem as variational problem in the space H¹(rot). This formulation has a comparatively simple structure and is closed to Maxwell's equations. The arising difficulty is that special finite elements, the Nedelec elements, must be used. The second approach circumvents this difficulty by employing a vector potential formulation. However, the structure of the eigenproblem becomes more difficult. It is an aim of the project to decide which of the variational formulations is more suitable for the numerical computation of eigensolutions.

• F. Schmidt, T. Friese, L. Zschiedrich, P. Deuflhard
  Adaptive Multigrid Methods for the Vectorial Maxwell Eigenvalue Problem for Optical Waveguide Design
  In: Mathematics - Key Technology for the Future: Joint Problems between Universities and Industry,  W. Jäger et al.  (ed) Springer,   270-292 (2003)
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• T.  Friese, P.  Deuflhard, F.  Schmidt
  A Multigrid Method for the Complex Helmholtz Eigenvalue Problem
  In: Procs. Eleventh International Conference on Domain Decomposition Methods in Sciences and Engineering (DD11), UK,  C.-H. Lai and P. E. Bj\orstad and M. Cross and O. B. Widlund  (ed) DDM.org,   19-26  -Available as ZIB preprint (SC 97-55) (1999)
  Abstract   Preprint  
• P. Deuflhard, T. Friese, F. Schmidt
  A Nonlinear Multigrid Eigenproblem Solver for the Complex Helmholtz Equation
   (SC 97-55) Konrad-Zuse-Zentrum Berlin  Preprint  -Appeared under the title: "A Multigrid Method for the Complex Helmholtz Eigenvalue Problem" in: Choi-Hong Lai et al. (eds.) Proc. 11th Int. Conf. on Domain Decomposition Methods. Bergen: DDM-org Press 1999. Pp. 18-26 (1997)
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• P. Deuflhard, T. Friese, F. Schmidt, R. März, H.-P. Nolting
  Effiziente Eigenmodenberechnung für den Entwurf integriert-optischer Chips
  In: Mathematik - Schlüsseltechnologie für die Zukunft,  Hoffmann,K. -H. et al.  (ed) Springer Verlag,   267-279 (1996)
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