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The problem under consideration is the wave equation for the electric field in a 3D cavity with perfectly conducting walls. When treated in the time domain an implicit timestepping is highly desirable due to its unconditional stability. In a finite-element setting each timestep involves the solution of a discrete variational problem for the bilinear form $(\cdot,\cdot)_0+(${\bf curl}$\cdot,$ {\bf curl}$\cdot)_0$ posed over {\bf H(curl; $\Omega$)}. I relied on Nedelec's {\bf H(curl; $\Omega$)}-conforming finite elements (edge elements), which yield a viable discretization for Maxwell's equations.
A multigrid method is employed as a fast iterative solution method. Since proper ellipticity of the bilinear form is confined to the complement of the kernel of the {\bf curl}--operator, discrete Helmholtz--decompositions of the finite element spaces are crucial for the design and analysis of the multigrid scheme.
Under certain assumptions on the computational domain and material functions, a rigorous proof of asymptotic optimality of the multigrid method can be given; it shows that convergence does not deteriorate on very fine grids. The results of numerical experiments confirm the practical efficiency of the method.
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