A robust and adaptive GenEO-type domain decomposition preconditioner for $\mathbf{H}(\mathbf{curl})$ problems in three-dimensional general topologies

Abstract: In this paper we design, analyse and test domain decomposition methods for linear systems of equations arising from conforming finite element discretisations of positive Maxwell-type equations, namely for $\mathbf{H}(\mathbf{curl})$ problems. It is well known that convergence of domain decomposition methods rely heavily on the efficiency of the coarse space used in the second level. We design adaptive coarse spaces that complement a near-kernel space made from the gradient of scalar functions. The new class of preconditioner is inspired by the idea of subspace decomposition, but based on spectral coarse spaces, and is specially designed for curl-conforming discretisations of Maxwell's equations in heterogeneous media on general domains which may have holes. We also address the practical robustness of various solvers in the case of non-trivial topologies and/or high aspect ratio of the domain.