The auxiliary space preconditioner for the de Rham complex

TL;DR

This paper extends auxiliary space preconditioners to n-dimensional de Rham complexes, providing theoretical analysis, implementation details, and numerical experiments demonstrating their effectiveness and scalability in four dimensions.

Contribution

It generalizes auxiliary space preconditioners to higher-dimensional de Rham complexes and develops a practical implementation for 4D cases using finite element exterior calculus.

Findings

Preconditioners are effective and scalable in 4D.

Numerical results confirm theoretical robustness.

Implementation leverages algebraic form operations.

Abstract

We generalize the construction and analysis of auxiliary space preconditioners to the n-dimensional finite element subcomplex of the de Rham complex. These preconditioners are based on a generalization of a decomposition of Sobolev space functions into a regular part and a potential. A discrete version is easily established using the tools of finite element exterior calculus. We then discuss the four-dimensional de Rham complex in detail. By identifying forms in four dimensions (4D) with simple proxies, form operations are written out in terms of familiar algebraic operations on matrices, vectors, and scalars. This provides the basis for our implementation of the preconditioners in 4D. Extensive numerical experiments illustrate their performance, practical scalability, and parameter robustness, all in accordance with the theory.