Multilevel solution of the time-harmonic Maxwellʹs equations based on edge elements

A widely used approach for the computation of time-harmonic electromagnetic elds is based on the well- known double-curl equation for either E or H, where edge elements are an appealing choice for nite element discretizations. Yet, the nullspace of the curl-operator comprises a considerable part of all spectral modes on the nite element grid. Thus standard multilevel solvers are rendered ine cient, as they essentially hinge on smoothing procedures like Gauss{Seidel relaxation, which cannot provide a satisfactory error reduction for modes with small or even negative eigenvalues. We propose to remedy this situation by an extended multilevel algorithm which relies on corrections in the space of discrete scalar potentials. After every standard V-cycle with respect to the canonical basis of edge elements, error components in the nullspace are removed by an additional projection step. Furthermore, a simple criterion for the coarsest mesh is derived to guarantee both stability and e ciency of the iterative multilevel solver. For the whole scheme we observe convergence rates independent of the re nement level of the mesh. The sequence of nested meshes required for our multilevel techniques is constructed by adaptive re nement. To this end we have devised an a posteriori error indicator based on stress recovery.