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In this paper we are concerned with the efficient solution of discrete variational problems related to the bilinear form ({\bf curl}\! , {\bf curl}\! $\cdot)_{{\font size{6}{0pt}\selectfont\textbf{\textit{L}}}^2(\Omega)}$ + ($\cdot,\cdot)_{{\font size{6}{0pt}\selectfont\textbf{\textit{L}}}^2(\Omega)}$ defined on \textbf{\textit{H}}({\bf curl}; ). This is a core task in the time-domain simulation of electromagnetic fields, if implicit timestepping is employed. We rely on Nédélec's \textbf{\textit{H}}({\bf curl}; )-conforming finite elements (edge elements) to discretize the problem.

We construct a multigrid method for the fast iterative solution of the resulting linear system of equations. Since proper ellipticity of the bilinear form is confined to the complement of the kernel of the {\bf curl} operator, Helmholtz decompositions are the key to the design of the algorithm: ({\bf curl}) and its complement ({\bf curl}) require separate treatment. Both can be …