Geometric multigrid preconditioners for unfitted discretisations of the de Rham complex

Abstract: We propose two families of multigrid preconditioners for unfitted finite element discretisations of the Riesz maps of the de Rham complex, in H(grad), H(curl) and H(div), stabilised by cell aggregation. The first is a geometric multigrid method applied directly to the unfitted operators, on a hierarchy of background meshes that share the same boundary representation. Its smoothers are star patches built on the aggregated mesh, so that each patch captures the kernel of the stabilisation. The second is a multilevel auxiliary-space preconditioner that combines a patch smoother localised around the cut cells with the body-fitted problem on the active mesh, which is in turn solved by a non-nested geometric multigrid method. Both are used as building blocks of block-diagonal preconditioners for the mixed Hodge Laplacians. Numerical experiments in two and three dimensions, on geometries with a controlled cut fraction and on general geometries, show iteration counts that are robust with respect to the cut configuration, the mesh size and the number of levels. We also report results for higher-order discretisations, weakly imposed Dirichlet boundary conditions and a face-based stabilisation.