For several PDEs naturally posed in the de Rham complex, standard (structure-preserving) variational discretizations lead to linear systems that can be indefinite (e.g., in mixed or saddle-point form), so classical point smoothers may be ineffective for multigrid. This situation arises, for example, for the Hodge-Dirac operator, mixed Hodge-Laplacians, and Maxwell-type formulations.
We propose a multigrid preconditioning framework that combines mass-lumping and transforming smoothers. On the discretization side, we replace the consistent FEEC (finite element exterior calculus) mass matrices by explicitly invertible mass-lumped mass matrices. On the solver side, we apply transforming smoothers that map the operator to a block form with positive definite diagonal blocks, enabling (distributive) Gauss-Seidel relaxation. Under mild $h$-uniform norm-equivalence assumptions, we prove spectral equivalence of the FEEC system and the mass-lumped one in the case of a trivial topology. This motivates using the mass-lumped MG cycles as preconditioners for the consistent FEEC system. The uniform convergence of the resulting multigrid cycles remains an open theoretical question.
We instantiate the approach for the Hodge-Dirac problem, the mixed Hodge-Laplacian, and a saddle-point system arising from magnetostatics. Extensive numerical experiments in MFEM validate the robustness of this preconditioner, demonstrating efficient V- and W-cycles in 2D/3D, and mesh-width-independent preconditioned GMRES iteration counts for the FEEC problem on unstructured meshes.