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Solving discrete constrained problems on de Rham complex

Zhongjie Lu

TL;DR

This work tackles discrete constrained problems on the de Rham complex, whose naive discretizations often yield ill-conditioned or non-invertible systems. By exploiting the de Rham complex structure, the authors transform the constrained problems into Laplace-like equivalents and develop a suite of equivalent formulations that decouple the constrained component and render the problems amenable to classical iterative methods and preconditioning. The approach is instantiated in FEEC-based discretizations, yielding discrete Hodge Laplacian problems that preserve a key $\mathcal{A}\mathcal{M}^{-1}\mathcal{B}=0$ property and admit efficient solution strategies regardless of inf-sup stability. The paper provides detailed theory for both $\mathcal{G}=0$ and $\mathcal{G}\neq 0$ cases, and supports the framework with extensive 3D numerical experiments for constrained Maxwell and grad-div problems, demonstrating robust convergence using ILU(0) preconditioning and eigen-solvers like LOBPCG. The results indicate that transforming to Laplace-like equivalent problems offers practical, scalable remedies for large-scale constrained discrete problems in electromagnetism and related areas.

Abstract

The main difficulty in solving the discrete constrained problem is its poor and even ill condition. In this paper, we transform the discrete constrained problems on de Rham complex to Laplace-like problems. This transformation not only make the constrained problems solvable, but also make it easy to use the existing iterative methods and preconditioning techniques to solving large-scale discrete constrained problems.

Solving discrete constrained problems on de Rham complex

TL;DR

This work tackles discrete constrained problems on the de Rham complex, whose naive discretizations often yield ill-conditioned or non-invertible systems. By exploiting the de Rham complex structure, the authors transform the constrained problems into Laplace-like equivalents and develop a suite of equivalent formulations that decouple the constrained component and render the problems amenable to classical iterative methods and preconditioning. The approach is instantiated in FEEC-based discretizations, yielding discrete Hodge Laplacian problems that preserve a key property and admit efficient solution strategies regardless of inf-sup stability. The paper provides detailed theory for both and cases, and supports the framework with extensive 3D numerical experiments for constrained Maxwell and grad-div problems, demonstrating robust convergence using ILU(0) preconditioning and eigen-solvers like LOBPCG. The results indicate that transforming to Laplace-like equivalent problems offers practical, scalable remedies for large-scale constrained discrete problems in electromagnetism and related areas.

Abstract

The main difficulty in solving the discrete constrained problem is its poor and even ill condition. In this paper, we transform the discrete constrained problems on de Rham complex to Laplace-like problems. This transformation not only make the constrained problems solvable, but also make it easy to use the existing iterative methods and preconditioning techniques to solving large-scale discrete constrained problems.

Paper Structure

This paper contains 25 sections, 10 theorems, 194 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Matrix aspects of the constrained problem
  3. The case $\mathcal{G} = 0$
  4. The case $\dim \mathbb{C}_0 = 0$
  5. The case $\dim \mathbb{C}_0 \not = 0$ and $c>0$
  6. The case $\dim \mathbb{C}_0 \not= 0$ and $c = 0$
  7. Summary
  8. The case $\mathcal{G} \not = 0$
  9. The discretization using finite element complexes
  10. The weak formulation
  11. The approximation for the mixed formulation
  12. The matrix forms of the discrete problem
  13. The discrete Hodge Laplacian problem
  14. Some aspects before numerical experiments
  15. How to solve the equivalent problems
  16. ...and 10 more sections

Key Result

Theorem 2.4

Figures (7)

  • Figure 1: The three 3D domains. $\Omega_1$ is a cube $\left[ 0 \;,\; \pi \right]^3$. $\Omega_2$ is the cube with a tunnel $\left[ \frac{\pi}{4} \;,\; \frac{3\pi}{4} \right]^2 \times \left[ 0 \;,\; \pi \right]$. $\Omega_3$ is the cube with a void $\left[ \frac{\pi}{4} \;,\; \frac{3\pi}{4} \right]^3$ inside.
  • Figure 2: The relative error of the solution $u_\varepsilon$ of penalty method \ref{['penalty_vare']} for the discrete problem of \ref{['example_equ_1']} with the exact solution.
  • Figure 3: The convergence histories of the equivalent problem \ref{['equiv_H0_c0_1_g1']} (left) and \ref{['equiv_H0_c0_2_g1']} (right) in solving the discrete problems of the equation \ref{['example_equ_1']}.
  • Figure 4: The convergence histories of the equivalent problem \ref{['equiv_H0_c0_1_g1']} (left) and \ref{['equiv_H0_c0_2_g1']} (right) in solving the discrete problems of the equation \ref{['example_equ_2']}.
  • Figure 5: The convergence histories of the eigenvalue problem (left) and linear systems (right) of the equivalent problem \ref{['equiv_H1_all_g1']} in solving the discrete problem of the equation \ref{['example_equ_3']}.
  • ...and 2 more figures

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • proof
  • Theorem 2.7
  • proof
  • ...and 16 more