Algebraic multigrid for high-order hierarchical H(curl) finite elements

Classic multigrid methods are often not directly applicable to nonelliptic problems such as curl-type partial differential equations (PDEs). Curl-curl PDEs require specialized smoothers that are compatible with the gradient-like (near) null space. Moreover, recent developments have focused on replicating the grad-curl-div de Rham complex in a multilevel hierarchy through smoothed aggregation based algebraic multigrid. These approaches have been successful for Nédélec finite elements (i.e., H(curl) edge elements), but do not extend naturally to high-order representations. In this paper we consider hierarchical high-order Whitney elements for the curl-curl eddy current problem and devise a scalable multilevel approach. Our method generates a hierarchy similar to p-type multigrid, which results in a coarse level that is amenable to further coarsening through the established process of a multilevel complex. The natural hierarchy of the elements induces an effective interpolation operator and motivates the construction of a compatible gradient smoothing process. We detail the multilevel solver for a hierarchical H(curl) basis and present numerical results in support of the method.