Multilevel Method for Mixed Eigenproblems

• Authors: R. Hiptmair;K. Neymeyr
• Venue: SIAM Journal on Scientific Computing
• Year: 2001

For a Lipschitz-polyhedron $\Omega\subset\mathbb{R}^3$ we consider eigenvalue problems $\operatorname{\bf curl}\alpha\operatorname{\bf curl}\mathbf{u}=\lambda\mathbf{u}$ and $\operatorname{grad}\alpha\operatorname{div}\mathbf{u} = \lambda\mathbf{u}$, $\lambda0$, set in $\boldsymbol{H}(\operatorname{\bf curl};\Omega)$ and $\boldsymbol{H}(\operatorname{div};\Omega)$. They are discretized by means of the conforming finite elements introduced by Nédélec. The preconditioned inverse iteration in its subspace variant is adapted to these problems. A standard multigrid scheme serves as the preconditioner. The main challenge arises from the large kernels of the operators curl and div. However, thanks to the choice of finite element spaces these kernels have a direct representation through the gradients/rotations of discrete potentials. This makes it possible to use a multigrid iteration in potential space to obtain approximate projections onto the orthogonal complements of the kernels. There is ample evidence that this will lead to an asymptotically optimal method. Numerical experiments confirm the excellent performance of the method even on very fine grids.