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Adaptive Multigrid Methods for the Vectorial Maxwell Eigenvalue Problem for Optical Waveguide Design

Please always quote using this URN: urn:nbn:de:0297-zib-6228
  • This paper has been motivated by the need for a fast robust adaptive multigrid method to solve the vectorial Maxwell eigenvalue problem arising from the design of optical chips. Our nonlinear multigrid methods are based on a previous method for the scalar Helmholtz equation, which must be modified to cope with the null space of the Maxwell operator due to the divergence condition. We present two different approaches. First, we present a multigrid algorithm based on an edge element discretization of time-harmonic Maxwell's equations, including the divergence condition. Second, an explicit elimination of longitudinal magnetic components leads to a nodal discretization known to avoid discrete \emph{spurious modes} also and a vectorial eigenvalue problem, for which we present a multigrid solver. Numerical examples show that the edge element discretization clearly outperforms the nodal element approach.

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Metadaten
Author:Frank Schmidt, Tilmann Friese, Lin Zschiedrich, Peter Deuflhard
Document Type:ZIB-Report
Tag:Maxwell's equations; edge elements; eigenvalue problem; multigrid methods; optical chip design; waveguide
MSC-Classification:65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N25 Eigenvalue problems
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N55 Multigrid methods; domain decomposition
Date of first Publication:2000/12/19
Series (Serial Number):ZIB-Report (00-54)
ZIB-Reportnumber:00-54
Published in:Appeared in: W. Jäger et al. (eds.) Mathematics - Key Technology for the Future : Joint Projects between Universities and Industry. Springer 2003. Pp. 279 -292
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