Multigrid Computation of Maxwell Eigenvalues

We consider the problem of solving the discrete Maxwell eigenvalue

problem

,

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in a closed simply connected cavity . For

related

eigenvalue problems for symmetric second order elliptic operators,

efficient

iterative schemes for the computation of a couple of the smallest

eigenvalues/eigenvectors have been proposed [1]. They are based on a

preconditioned inverse iteration and a comprehensive analysis has been

presented in

[3].

In the case of the Maxwell eigenvalue problem the large kernel of the

-operator thwarts the straightforward appli

cation of

these algorithms.

However, when the discretization is based on edge elements, we have an

explicit

representation of through gradients of

linear finite

element functions. This paves the way for a fast approximate projection

onto

, which can be coupled with the

edge element

multigrid scheme developed by the author [2]. Numerical experiments

confirm the

good performance of this approach for large scale problems.

References

 

1

 

J. BRAMBLE, A. KNYAZEV, AND J. PASCIAK, A subspace

preconditioning

algorithm for eigenvector/eigenvalue computation, Advances Comp.

Math., 6

(1996), pp. 159-189.

2

 

R. HIPTMAIR, Multigrid method for Maxwell's equations,

SIAM J.

Numer. Anal., 36 (1999), pp. 204-225.

3