Solving discrete constrained problems on de Rham complex

TL;DR

This paper introduces a transformation of discrete constrained problems on de Rham complex into Laplace-like problems, improving solvability and enabling the use of existing iterative methods and preconditioning techniques for large-scale problems.

Contribution

It presents a novel transformation approach that converts difficult constrained problems into more manageable Laplace-like problems, facilitating their solution.

Findings

Transformation improves problem conditioning

Enables use of existing iterative solvers

Facilitates solving large-scale problems

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.