Multilevel solution of the time-harmonic Maxwell's equations based on edge elements

Abstract

A widely used approach for the computation of time-harmonic electromagnetic fields is based on the well-known double-curl equation for either E or H, where edge elements are an appealing choice for finite element discretizations. Yet, the nullspace of the curl-operator comprises a considerable part of all spectral modes on the finite element grid. Thus standard multilevel solvers are rendered inefficient, 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 efficiency of the iterative multilevel solver. For the whole scheme we observe convergence rates independent of the refinement level of the mesh. The sequence of nested meshes required for our multilevel techniques is constructed by adaptive refinement. To this end we have devised an a posteriori error indicator based on stress recovery. Copyright © 1999 John Wiley & Sons, Ltd.