Multigrid Method for Maxwell's Equations

Abstract

In this paper we are concerned with the efficient solution of discrete variational problems related to the bilinear form ({\bf curl}\! $\cdot$, {\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}}$_0$({\bf curl}; $\Omega$). 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}; $\Omega$)-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: ${\cal N}$({\bf curl}) and its complement ${\cal N}$({\bf curl})$^{\bot}$ require separate treatment. Both can be tackled by nodal multilevel decompositions, where for the former the splitting is set in the space of discrete scalar potentials. Under certain assumptions on the computational domain and the material functions, a rigorous proof of the asymptotic optimality of the multigrid method can be given, which shows that convergence does not deteriorate on very fine grids. The results of numerical experiments confirm the practical efficiency of the method.