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.

翻译：$L^2$ de Rham复形的Riesz映射常作为更复杂问题快速预条件子构造中的子问题出现。本文针对这些Riesz映射的高阶有限元离散，提出了与求和分解算子应用具有相同时空复杂度的多重网格求解器——即在Krylov方法框架下实现多项式度的最优复杂度。我们的核心思想是为de Rham复形中各空间构造新型有限元，使其在参考六面体上的$L^2$和$H(\mathrm{d})$内积（$\mathrm{d} \in \{\mathrm{grad}, \mathrm{curl}, \mathrm{div}\}$）中均具有正交性。在可分离情形下，由此产生的稀疏性使得Pavarino、Arnold-Falk-Winther以及Hiptmair空间分解中的区域块问题能够被快速求解。对于不可分离情形，该方法可应用于通过构造即具有稀疏性的辅助算子。若采用精确Cholesky分解处理稀疏区域块问题，求解复杂度达到最优，但建立开销和存储需求并非最优。我们通过引入更精细的Hiptmair空间分解以及采用基于静态凝聚稀疏模式的非完全Cholesky分解来解决此问题——无论求解器是否使用静态凝聚，该模式均适用。由此产生的多重网格松弛方法在时间与空间复杂度上均实现多项式度的最优性。