# Multigrid Method for Maxwell's Equations

Tuesday, June 10, 1997 - 2:20pm - 2:40pm

Keller 3-180

Ralf Hiptmair (Universität Augsburg)

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.

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.