Auxiliary iterative schemes for the discrete operators on de Rham complex

TL;DR

This paper introduces auxiliary iterative schemes based on the Hodge Laplacian for efficiently solving discrete operators on de Rham complex, addressing challenges posed by their large kernels.

Contribution

It develops a new framework using auxiliary schemes with Laplace-like spectra to improve iterative solutions of discrete de Rham operators, including preconditioning strategies.

Findings

Spectra of new schemes are Laplace-like, enabling efficient iterative methods.

Preconditioners like ILU and Multigrid significantly improve convergence.

Numerical experiments confirm the efficiency and practicality of the proposed schemes.

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.