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Multigrid Preconditioning for FEEC using Mass-Lumping and Transforming Smoothers

by R. Dabetic

(Report number 2026-24)

Abstract
For PDEs naturally posed in the de Rham complex, structure-preserving mixed and saddle-point finite element discretizations typically produce indefinite linear systems. We propose a multigrid preconditioning framework that combines mass-lumped (explicitly invertible) FEEC mass matrices with transforming smoothers that map the operator to a block form with positive definite diagonal blocks, enabling Gauss-Seidel-type relaxation on the transformed system. Under mild h-uniform norm-equivalence assumptions (and for trivial topology), we prove stability of the mass-lumped systems, and by extension spectral equivalence between the mass-lumped and original FEEC operators, which motivates using multigrid cycles designed for the mass-lumped operators as preconditioners for the consistent FEEC systems. While our primary focus is on algorithmic design rather than formal convergence theory, extensive numerical experiments on the Hodge-Dirac operator, mixed Hodge-Laplacians, and a magnetostatics saddle-point system in 2D and 3D demonstrate the robustness of the approach.

Keywords: Multigrid, Mixed Finite Elements, Saddle-Point Systems, Transforming Smoothers, Distributive Relaxation, Hodge-Dirac Operator, Hodge-Laplacian, Maxwell's Equations, Magnetostatics, Boundary Value Problem, FEEC, DEC

@Techreport{D26_1179,
  author = {R. Dabetic},
  title = {Multigrid Preconditioning for FEEC using Mass-Lumping and Transforming Smoothers},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-24},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-24.pdf },
  year = {2026}
}

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