Analysis of multilevel methods for eddy current problems

In papers by Arnold, Falk, and Winther, and by Hiptmair, novel multigrid methods for discrete H(curl; Omega)-elliptic boundary value problems have been proposed. Such problems frequently occur in computational electromagnetism, particularly in the context of eddy current simulation. This paper focuses on the analysis of those nodal multilevel decompositions of the spaces of edge finite elements that form the foundation of the multigrid methods. It provides a significant extension of the existing theory to the case of locally vanishing coefficients and nonconvex domains. In particular, asymptotically uniform convergence of the multigrid method with respect to the number of refinement levels can be established under assumptions that are satisfied in realistic settings for eddy current problems. The principal idea is to use approximate Helmholtz-decompositions of the function space H(curl; Omega) into an H-1(Omega)-regular subspace and gradients. The main results of standard multilevel theory for H-1(Omega)-elliptic problems can then be applied to both subspaces. This yields preliminary decompositions still outside the edge element spaces. Judicious alterations can cure this.