Fast solvers for the high-order FEM simplicial de Rham complex: Extended edition

TL;DR

This paper introduces high-order finite element methods with specialized basis functions for efficiently solving Riesz maps in the de Rham complex, significantly reducing computational complexity in three dimensions.

Contribution

The authors develop new finite elements with orthogonal basis functions that enable p-robust iterative solutions and efficient preconditioning for high-order de Rham complex problems.

Findings

Achieved p-robust solution of Riesz maps with $ ext{O}(p^6)$ flops in 3D.

Designed a preconditioning strategy by neglecting weak interior couplings.

Applied methods to solve Hodge Laplacians with novel augmented Lagrangian preconditioners.

Abstract

We present new finite elements for solving the Riesz maps of the de Rham complex on triangular and tetrahedral meshes at high order. The finite elements discretize the same spaces as usual, but with different basis functions, so that the resulting matrices have desirable properties. These properties mean that we can solve the Riesz maps to a given accuracy in a $p$-robust number of iterations with $\mathcal{O}(p^6)$ flops in three dimensions, rather than the na\"ive $\mathcal{O}(p^9)$ flops. The degrees of freedom build upon an idea of Demkowicz et al., and consist of integral moments on an equilateral reference simplex with respect to a numerically computed polynomial basis that is orthogonal in two different inner products. As a result, the interior-interface and interior-interior couplings are provably weak, and we devise a preconditioning strategy by neglecting them. The combination of this approach with a space decomposition method on vertex and edge star patches allows us to efficiently solve the canonical Riesz maps at high order. We apply this to solving the Hodge Laplacians of the de Rham complex with novel augmented Lagrangian preconditioners.