Multigrid solvers for the de Rham complex with optimal complexity in polynomial degree

TL;DR

This paper develops multigrid solvers for high-order finite element discretizations of the Riesz maps in the de Rham complex, achieving optimal complexity in polynomial degree through novel finite elements with special orthogonality properties.

Contribution

It introduces new finite elements with orthogonality in both $L^2$ and $H(d)$ inner products, enabling optimal multigrid solvers for Riesz maps with high polynomial degrees.

Findings

Achieves optimal complexity in polynomial degree for multigrid solvers.

Uses new finite elements with orthogonality properties for efficient patch problem solutions.

Employs incomplete Cholesky factorizations to reduce setup costs and storage.

Abstract

The Riesz maps of the $L^2$ de Rham complex frequently arise as subproblems in the construction of fast preconditioners for more complicated problems. In this work we present multigrid solvers for high-order finite element discretizations of these Riesz maps with the same time and space complexity as sum-factorized operator application, i.e.~with optimal complexity in polynomial degree in the context of Krylov methods. The key idea of our approach is to build new finite elements for each space in the de Rham complex with orthogonality properties in both the $L^2$- and $H(\mathrm{d})$-inner products ($\mathrm{d} \in \{\mathrm{grad}, \mathrm{curl}, \mathrm{div}\})$ on the reference hexahedron. The resulting sparsity enables the fast solution of the patch problems arising in the Pavarino, Arnold--Falk--Winther and Hiptmair space decompositions, in the separable case. In the non-separable case, the method can be applied to an auxiliary operator that is sparse by construction. With exact Cholesky factorizations of the sparse patch problems, the application complexity is optimal but the setup costs and storage are not. We overcome this with the finer Hiptmair space decomposition and the use of incomplete Cholesky factorizations imposing the sparsity pattern arising from static condensation, which applies whether static condensation is used for the solver or not. This yields multigrid relaxations with time and space complexity that are both optimal in the polynomial degree.