Algebraic Multigrid for Edge Elements by Component Splitting

Rudolf Beck

Abstract:

This talk is about the solution of Maxwell's equations in the low frequency range; the applications we aim at are implicit time-stepping schemes for eddy current computations and the stationary double-curl equation for time-harmonic fields. We assume that the computational domain is discretized by Nédélec's H(curl)-conforming edge elements of the lowest order.

Algebraic coarsening strategies for such finite element spaces are harder to devise than in the case of Lagrange-type elements. The difficulties are basically due to the geometric structure of the vectorial shape functions.

The solution presented here relies on a spatial component splitting of the fields, where mesh coarsening takes place in an auxiliary subspace constructed with the aid of Lagrange-type basis functions. Within this subspace coarse grids are created recursively by an advancing-front algorithm, which merely exploits the matrix graphs.

Unfortunately, the non-trivial kernel of the curl-operator cannot be tackled within this setting. Thus we resort to a discrete Helmholtz decomposition of the fields, allowing an efficient smoothing of the kernel modes by a separate algebraic multigrid cycle.

Some numerical experiments will be presented in order to assess the efficacy of the proposed algorithms.