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Analysis of injection operators in multigrid solvers for hybridized discontinuous Galerkin methods

Peipei Lu, Andreas Rupp, Guido Kanschat

TL;DR

The paper addresses the efficient solution of HDG discretizations for diffusion problems by proving uniform convergence of a geometric multigrid V-cycle under new, locality-friendly injection-operator assumptions. It develops an abstract injection-operator framework and identifies several strictly local options that maintain stability and enable quasi-orthogonality, while relaxing stabilization requirements to τ_ℓ h_ℓ ≲ 1. The main contributions include establishing energy-stability and Ritz-projection bounds, proving a finite-set of preconditions that guarantee convergence, and showing that the analysis extends to HDG variants RT-H and BDM-H (and LDG-H) through LS1–LS6. Numerical experiments corroborate the theory, demonstrating robust multigrid performance across mesh refinements, polynomial degrees, and different injection operators. This work advances practical, scalable multigrid solvers for HDG methods by enabling local injections and broader parameter regimes without sacrificing convergence guarantees.

Abstract

Uniform convergence of the geometric multigrid V-cycle is proven for HDG methods with a new set of assumptions on the injection operators from coarser to finer meshes. The scheme involves standard smoothers and local solvers which are bounded, convergent, and consistent. Elliptic regularity is used in the proofs. The new assumptions admit injection operators local to a single coarse grid cell. Examples for admissible injection operators are given. The analysis applies to the hybridized local discontinuous Galerkin method, hybridized Raviart-Thomas, and hybridized Brezzi-Douglas-Marini mixed element methods. Numerical experiments are provided to confirm the theoretical results.

Analysis of injection operators in multigrid solvers for hybridized discontinuous Galerkin methods

TL;DR

The paper addresses the efficient solution of HDG discretizations for diffusion problems by proving uniform convergence of a geometric multigrid V-cycle under new, locality-friendly injection-operator assumptions. It develops an abstract injection-operator framework and identifies several strictly local options that maintain stability and enable quasi-orthogonality, while relaxing stabilization requirements to τ_ℓ h_ℓ ≲ 1. The main contributions include establishing energy-stability and Ritz-projection bounds, proving a finite-set of preconditions that guarantee convergence, and showing that the analysis extends to HDG variants RT-H and BDM-H (and LDG-H) through LS1–LS6. Numerical experiments corroborate the theory, demonstrating robust multigrid performance across mesh refinements, polynomial degrees, and different injection operators. This work advances practical, scalable multigrid solvers for HDG methods by enabling local injections and broader parameter regimes without sacrificing convergence guarantees.

Abstract

Uniform convergence of the geometric multigrid V-cycle is proven for HDG methods with a new set of assumptions on the injection operators from coarser to finer meshes. The scheme involves standard smoothers and local solvers which are bounded, convergent, and consistent. Elliptic regularity is used in the proofs. The new assumptions admit injection operators local to a single coarse grid cell. Examples for admissible injection operators are given. The analysis applies to the hybridized local discontinuous Galerkin method, hybridized Raviart-Thomas, and hybridized Brezzi-Douglas-Marini mixed element methods. Numerical experiments are provided to confirm the theoretical results.

Paper Structure

This paper contains 21 sections, 11 theorems, 91 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Model equation and discretization
  3. Spaces for the HDG multigrid method
  4. Hybrid discontinuous Galerkin method for the diffusion equation
  5. Operators for the multigrid method and analysis
  6. Assumptions on local solvers
  7. Injection operators
  8. Assumptions on injection operators
  9. Possible injection operators
  10. Multigrid method and main convergence result
  11. Multigrid algorithm
  12. Main convergence result
  13. Convergence analysis
  14. Energy stability of the injection and proof of \ref{['EQ:precond2']}
  15. Proof of \ref{['EQ:precond1']}
  16. ...and 6 more sections

Key Result

Lemma 3.1

\newlabelLEM:quasi_orth0 Assuming EQ:IA2 and EQ:LS4, then for any $\lambda \in M_{\ell}$ there holds:

Figures (2)

  • Figure 1: Examples of original faces (blue) faces of refined mesh (red) for regular refinement (left) and bisection (right). Faces which are refinements are solid and newly introduced faces are dashed.
  • Figure 1: Initial mesh for numerical experiments.

Theorems & Definitions (22)

  • Remark 2.1
  • Lemma 3.1: Quasi-orthogonality
  • Proof 1
  • Lemma 3.2
  • Proof 2
  • Theorem 4.1
  • Lemma 5.1
  • Lemma 5.2
  • Proof 3
  • Lemma 5.3
  • ...and 12 more