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.
From: Radovan Dabetic [view email]
[v1]
Tue, 16 Jun 2026 13:43:11 UTC (7,519 KB)
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