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Auxiliary iterative schemes for the discrete operators on de Rham complex

Zhongjie Lu

TL;DR

This work tackles the computational challenge of solving discrete $d^*d$ problems on the de Rham complex, where large kernels hinder iterative solvers. It introduces auxiliary schemes built from the full Hodge Laplacian to produce Laplace-like spectra, enabling the use of standard Laplace-based solvers and preconditioners; after solving the auxiliary problems, the original $d^*d$ solutions are recovered via simple corrections and recomputed eigenvalues. The authors provide matrix realizations, propose replacing dense mass inverses with a sparse SPD operator to maintain sparsity, and propose ILU-type, SAIT, and multigrid preconditioners. They apply the framework to 3D Maxwell and grad-div problems, showing improved convergence, reliable eigenvalue recovery, and scalable performance on two challenging domains. The results indicate that the auxiliary schemes offer a practical pathway to efficient iterative solution of hiddenly ill-posed de Rham operators in finite element contexts.

Abstract

The main difficulty in solving the discrete source or eigenvalue problems of the operator $ d^*d $ with iterative methods is to deal with its huge kernel, for example, the $ \nabla \times \nabla \times $ and $- \nabla ( \nabla \cdot ) $ operator. In this paper, we construct a kind of auxiliary schemes for their discrete systems based on Hodge Laplacian on de Rham complex. The spectra of the new schemes are Laplace-like. Then many efficient iterative methods and preconditioning techniques can be applied to them. After getting the solutions of the auxiliary schemes, the desired solutions of the original systems can be recovered or recognized through some simple operations. We sum these up as a new framework to compute the discrete source and eigenvalue problems of the operator $ d^*d $ using iterative method. We also investigate two preconditioners for the auxiliary schemes, ILU-type method and Multigrid method. Finally, we present plenty of numerical experiments to verify the efficiency of the auxiliary schemes.

Auxiliary iterative schemes for the discrete operators on de Rham complex

TL;DR

This work tackles the computational challenge of solving discrete problems on the de Rham complex, where large kernels hinder iterative solvers. It introduces auxiliary schemes built from the full Hodge Laplacian to produce Laplace-like spectra, enabling the use of standard Laplace-based solvers and preconditioners; after solving the auxiliary problems, the original solutions are recovered via simple corrections and recomputed eigenvalues. The authors provide matrix realizations, propose replacing dense mass inverses with a sparse SPD operator to maintain sparsity, and propose ILU-type, SAIT, and multigrid preconditioners. They apply the framework to 3D Maxwell and grad-div problems, showing improved convergence, reliable eigenvalue recovery, and scalable performance on two challenging domains. The results indicate that the auxiliary schemes offer a practical pathway to efficient iterative solution of hiddenly ill-posed de Rham operators in finite element contexts.

Abstract

The main difficulty in solving the discrete source or eigenvalue problems of the operator with iterative methods is to deal with its huge kernel, for example, the and operator. In this paper, we construct a kind of auxiliary schemes for their discrete systems based on Hodge Laplacian on de Rham complex. The spectra of the new schemes are Laplace-like. Then many efficient iterative methods and preconditioning techniques can be applied to them. After getting the solutions of the auxiliary schemes, the desired solutions of the original systems can be recovered or recognized through some simple operations. We sum these up as a new framework to compute the discrete source and eigenvalue problems of the operator using iterative method. We also investigate two preconditioners for the auxiliary schemes, ILU-type method and Multigrid method. Finally, we present plenty of numerical experiments to verify the efficiency of the auxiliary schemes.

Paper Structure

This paper contains 20 sections, 8 theorems, 106 equations, 7 figures, 14 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The abstract Hodge Laplacian and its approximation
  3. The abstract Hodge Laplacian
  4. The approximation for Hodge Laplacian
  5. The analysis of discrete operators
  6. Auxiliary schemes for the discrete $d^*d$ problems
  7. The auxiliary discrete source problem
  8. The auxiliary discrete eigenvalue problem
  9. Why do we use the auxiliary formulations?
  10. The matrix forms of the auxiliary schemes
  11. The modification of $\mathcal{B}^T \mathcal{M}_{k-1}^{-1}\mathcal{B}$
  12. Auxiliary scheme for the source problem
  13. Auxiliary scheme for the eigenvalue problem
  14. Two preconditioners
  15. Incomplete LU factorization and sparse approximate inverse preconditioner
  16. ...and 5 more sections

Key Result

Lemma 3.1

We have the orthogonal decompositions for $V^k_h$:

Figures (7)

  • Figure 1: Left: a cubic domain $[0,\pi]^3$. Right: a cubic domain with a hole $[0,\pi]^3 \setminus [\pi/4, 3\pi/4]^2 \times [0,\pi]$.
  • Figure 2: The number of the nonzero entries in the SAIT matrices for the auxiliary discrete Maxwell source problems. Left: Domain 1. Right: Domain 2.
  • Figure 3: The convergence histories of the original discrete Maxwell equation and its auxiliary scheme with different iterative methods. The mesh is level 5 with the mesh size $h = \pi/64$. Left: Domain 1. Rigth: Domain 2.
  • Figure 4: The convergence histories of the mass equation. The relative error is computed in two ways: by the mass equation and by the original discrete Maxwell equation. Left: Domain 1. Rigth: Domain 2.
  • Figure 5: Compute the first 20 eigenvalue of the auxiliary discrete Maxwell operator on Domain 1 with the mesh of level 5 ($h = \frac{\pi}{64}$) using LOBPCG method with ILU(0), SAIT$(0.05,10)$, Multigrid preconditioner, respectively.
  • ...and 2 more figures

Theorems & Definitions (16)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • Theorem 5.2
  • proof
  • Theorem 5.3
  • ...and 6 more