Multigrid Preconditioning for FEEC using Mass-Lumping and Transforming Smoothers

Technical Summary: Multigrid Preconditioning for FEEC using Mass-Lumping and Transforming Smoothers

Problem Statement
The paper addresses the challenge of solving linear systems arising from Finite Element Exterior Calculus (FEEC) discretizations of partial differential equations (PDEs) posed on the de Rham complex. Specifically, it targets mixed and saddle-point formulations for the Hodge-Dirac operator, the Hodge-Laplacian, and magnetostatic systems (Maxwell's equations in the Coulomb gauge). These discretizations typically yield indefinite linear systems with saddle-point structures. Standard iterative smoothers, such as Gauss-Seidel or Jacobi, are ineffective for these indefinite systems because the operators lack the positive definiteness required for such relaxation schemes. While previous work has proposed multigrid approaches based on Schur complements, this paper seeks an alternative strategy that avoids the complexity of Schur complement inversion.

Methodology
The proposed framework combines mass-lumping with transforming smoothers to construct an effective preconditioner for the consistent FEEC systems.

1. Mass-Lumping and Spectral Equivalence:
   The authors introduce a mass-lumped version of the FEEC operators where the consistent mass matrices are replaced by diagonal (or explicitly invertible) matrices. This allows for the explicit inversion of the Riesz isomorphism, a step that is otherwise prohibitively expensive with dense consistent mass matrices.

   • Under mild $h$-uniform norm-equivalence assumptions between the consistent and mass-lumped inner products, the paper proves a discrete Poincaré inequality for the mass-lumped norms.
   • This stability result leads to a proof of spectral equivalence between the original consistent FEEC operators and the mass-lumped operators. This equivalence justifies using the mass-lumped system as a proxy for preconditioning the original system.

2. Transforming Smoothers:
   To handle the indefiniteness of the mass-lumped operators, the method employs transforming smoothers. The core idea is to algebraically transform the operator into a block form where the diagonal blocks are positive definite, enabling the use of standard smoothers like Gauss-Seidel.

   • For the Hodge-Dirac, Hodge-Laplacian, and magnetostatics problems, specific left- and right-transformation matrices ($T_L, T_R$) are constructed.
   • These transformations map the original indefinite system to a lower-block-triangular system (or a system with positive definite diagonal blocks).
   • A Gauss-Seidel relaxation is applied to this transformed system, and the result is mapped back to the original space.

3. Multigrid Preconditioning:
   The multigrid algorithm is designed specifically for the mass-lumped, transformed system. This multigrid cycle ($M_{MG}$) serves as an approximate inverse for the mass-lumped operator.

   • To solve the original FEEC system $G_h u = g$, the preconditioner $B$ is constructed as $B \approx J_h^{-1} M_{MG} \tilde{H}_h (J_h^*)^{-1}$, where $J_h$ is the isomorphism between the standard and mass-lumped spaces (often the identity in practice) and $\tilde{H}_h$ is the mass-lumped Riesz map.
   • Because the resulting preconditioned system remains indefinite, the authors utilize GMRES (Generalized Minimal Residual method) as the outer solver, as standard methods like PCG or MINRES require positive definite preconditioners.

Key Contributions

• Algorithmic Design: The paper proposes a unified multigrid preconditioning framework that integrates mass-lumping with transforming smoothers for FEEC problems.
• Theoretical Stability: It establishes the stability of the mass-lumped systems and proves spectral equivalence to the original FEEC operators under mild assumptions, providing a theoretical basis for the preconditioning strategy.
• Handling Indefiniteness: The work demonstrates how transforming smoothers can convert indefinite block operators into forms amenable to classical relaxation techniques without requiring Schur complement approximations.
• Robustness on Complex Topologies: The method is shown to handle domains with non-trivial topologies (harmonic forms) effectively, where the rank deficiency of the system is determined by the Betti numbers.

Numerical Results
Extensive numerical experiments were conducted in 2D and 3D on various meshes, including those with reentrant corners and holes (non-trivial topology).

• Convergence: The multigrid cycles (V-cycle and W-cycle) combined with the transforming smoothers demonstrated robust convergence. The spectral radii of the error propagation operators remained bounded away from unity, independent of the mesh size $h$.
• Mass-Lumping Strategies: Three mass-lumping strategies were tested: scaled identity, row sums, and barycentric duals (inspired by Discrete Exterior Calculus). The results indicated that these different strategies did not lead to pronounced qualitative differences in convergence behavior.
• Solver Performance: When used as a preconditioner for GMRES, the method yielded iteration counts that were largely independent of the mesh refinement level. The W-cycle generally provided more robust convergence than the V-cycle.
• Harmonic Forms: The experiments confirmed that the method correctly handles the rank deficiency associated with harmonic forms, with the number of non-convergent eigenmodes matching the dimension of the harmonic space.

Significance and Claims
The authors position this work primarily as an algorithmic contribution rather than a formal convergence theory proof. They explicitly state that while they prove spectral equivalence and stability, a rigorous convergence theory establishing uniform bounds on GMRES iteration counts remains an open problem for future investigation.

The significance of the approach lies in its ability to provide a practical, robust preconditioner for indefinite FEEC systems using explicit, cheap operations (mass-lumping) and algebraic transformations, avoiding the complexities of Schur complement methods. The numerical evidence suggests the approach is a valid and promising direction for solving mixed finite element problems in electromagnetism and related fields, warranting further theoretical study and extension to other problems like Stokes flow or systems with coefficient jumps.