Fitted norm preconditioners for the Hodge Laplacian in mixed form

TL;DR

This paper develops and analyzes a new class of norm-equivalent preconditioners for the Hodge Laplacian in mixed form, ensuring stability and efficiency in solving related saddle point problems numerically.

Contribution

It introduces a unified framework for constructing practical, norm-equivalent preconditioners for the Hodge Laplacian, improving stability and computational efficiency.

Findings

Preconditioners demonstrate fast convergence in numerical experiments.

Uniformly bounded iteration counts across different problem instances.

Applicable to 2D and 3D Hodge Laplace problems.

Abstract

We use the practical framework for abstract perturbed saddle point problems recently introduced by Hong et al. to analyze the mixed formulation of the Hodge Laplace problem. We compose two parameter-dependent norms in which the uniform continuity and stability of the problem follow. This not only guarantees the well-posedness of the corresponding variational formulation on the continuous level, but also of related compatible discrete models. We further simplify the obtained norms and, in both cases, arrive at the same norm-equivalent preconditioner that is easily implementable. The efficiency and uniformity of the preconditioner are demonstrated numerically by the fast convergence and uniformly bounded number of preconditioned MINRES iterations required to solve various instances of Hodge Laplace problems in two and three space dimensions.