Time-harmonic Maxwell equations in a lossless cavity lead to a second order differential equation for electric field involving a differential operator that is neither elliptic nor definite. A Galerkin method using Nedelec's curl-conforming finite elements can be employed to get approximate solutions numerically. In this talk, results of [4, 5] on the suitability of multigrid and Schwarz methods for efficiently solving the resulting indefinite linear system will be presented.

The analysis of the present work builds on techniques in [1, 6] to overcome the difficulties caused by the non-ellipticity of the operator. As in their analysis, discrete Helmholtz decompositions play a crucial role. A new and critical ingredient of our analysis is an estimate on discrete solution operators of Maxwell equations. This estimate allows analysis of overlapping Schwarz methods in the spirit of perturbation arguments in [3], even though our operator is not elliptic. It also allows analysis of a `\' multigrid cycle using techniques of [2], although prima facie it may appear that ellipticity is required for application of these techniques. In both cases, the analysis involves comparison of operators with their analogues in positive definite case.

Of practical importance is the fact that some algorithms that work well in the elliptic case, fails in our application. It has now been known for some time that a multigrid V-cycle with a point-Jacobi smoother is not appropriate for our problem. However, we show that a multigrid algorithm with block Jacobi or block Gauss-Seidel smoother, if the blocks are chosen as in [1], leads to good convergence rates, provided the coarse mesh used is sufficiently fine. Our results also indicate that, in contrast to the elliptic case, replacing subdomain solves by equivalent operations in overlapping Schwarz methods does not lead to good results.

References:

[1] D. N. Arnold, R. S. Falk and R. Winther. Multigrid in H(div) and H(curl). Numer. Math., 85(2):197-217, 2000.
[2] J. H. Bramble, D. Y. Kwak, and J. E. Pasciak. Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems. SIAM J. Numer. Anal., 31(6):1746-1763, 1994.
[3] X.-C. Cai and O. B. Widlund. Domain decomposition algorithms for indefinite elliptic problems. SIAM J. Sci. Stat. Comput., 13(1):243-258, 1992.
[5] J. Gopalakrishnan and J. E. Pasciak. Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell equations. Submitted. Available as IMA Preprint 1711, June 2000.
[6] J. Gopalakrishnan, J. E. Pasciak and L. Demkowicz. A multigrid algorithm for time harmonic Maxwell equations. In preparation.
[4] R. Hiptmair. Multigrid method for Maxwell's equations. SIAM J. Numer. Anal., 36(1):204-225, 1999.