Multilevel Solution of the Time-Harmonic Maxwell’s Equations Based on Edge Elements

A widely used approach for the computation of time-harmonic electromag-netic fields is based on the well-known double-curl equation for either E or H. An appealing choice for finite element discretizations are edge elements, the lowest order variant of a H(curl)-conforming family of finite elements. However, the large nullspace of the curl-operator gives rise to serious dif-ficulties. It comprises a considerable part of all spectral modes on the finite element grid, tending to pollute the solution with non-physical contributions and crippling standard multilevel solvers. We tackle these problems by an adaptive multilevel algorithm. After every standard V-cycle with respect to the canonical basis of edge elements, the non-physical contributions are removed by a projection step. It requires the solution of Poisson’s equation, augmented by certain boundary terms, in the nullspace, which can be carried out efficiently by an inner multilevel iteration. The whole scheme yields convergence rates independent of the refinement level of the mesh. Furthermore, a simple criterion for meshes is derived which can resolve all field modes corresponding to negative eigenvalues. This requirement is essential to guarantee both stability and efficiency of an iterative multilevel solver for indefinite systems. For controlling adaptive mesh refinement we have devised an a posteriori error indicator based on stress recovery. Numerical experiments demonstrate the efficiency of the method for the simulation of waveguides. Key words. Maxwell’s equations, edge elements, Nédélec elements, scattering problems, waveguide computation