Multigrid Solvers for the de Rham Complex with Optimal Complexity in Polynomial Degree

. The Riesz maps of the \ (L²\) 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 \ (L²\) - and \ (H (d) \) -inner products (\ (d \grad, curl, 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 nonseparable 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. Reproducibility of computational results. This paper has been awarded the "SIAM Reproducibility Badge: Code and data available" as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https: //doi. org/10. 5281/zenodo. 7358044 and in the supplementary materials (pmgdeᵣham. zip 61. 2KB). Keywordspreconditioningde Rham complexhigh-orderadditive Schwarzincomplete Choleskyoptimal complexityMSC codes65F0865N3565N55