Tensor-Mehrgitterverfahren für die zylindersymmetrische Maxwell-Gleichung

Many electrical devices like Resonators or Cables exhibit rotational symmetry. When simulating these devices, it is desirable to exploit the symmetry in order to reduce computational effort. The standard approach to this is the transformation to suitable coordinates and subsequent reduction to a two-dimensional problem.

The change of coordinates causes the coefficients of the two-dimensiona l problem to become singular, so standard multigrid techniques do not provide level-independent rates of convergence. To solve this problem, block smoothing procedures and semicoarsening techniques can be used.

For rectangular regions, we can prove convergence estimates that do not depend on the level of the finest grid or on the behaviour of the coefficients in one direction. In the proof, only estimates for one-dimensional Helmholtz problems are needed.

Numerical examples demonstrate that the algorithms work for non-rectangular regions, too.