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

Pablo D. Brubeck, Patrick E. Farrell, Robert C. Kirby, Charles Parker

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 naïve $\mathcal{O}(p^9)$ 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.

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

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 -robust number of iterations with flops in three dimensions, rather than the naïve 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.

Paper Structure

This paper contains 41 sections, 15 theorems, 136 equations, 8 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Finite element discretization
  3. Orthogonality-promoting discretizations
  4. Degrees of freedom with commuting diagram properties
  5. Local space decomposition
  6. Abstract construction of orthogonality-promoting basis
  7. Concrete definition of orthogonality-promoting degrees of freedom
  8. Degrees of freedom for $H(\mathop{\mathrm{grad}}\nolimits)$
  9. Degrees of freedom for $H(\mathop{\mathrm{curl}}\nolimits)$
  10. Degrees of freedom for $H(\mathop{\mathrm{div}}\nolimits)$
  11. Properties of the basis on the reference cell
  12. Relations to Whitney forms and eigenfunctions
  13. Algebraic properties
  14. Comparison to existing bases
  15. Properties of the global basis functions
  16. ...and 26 more sections

Key Result

Lemma 3.1

The degrees of freedom eq:abstract-dofs-stiff are unisolvent on $X^k(\hat{T})$.

Figures (8)

  • Figure 1: Flop counts, peak memory usage, and nonzeros in the sparse matrices and patch factors in the solution of the Riesz maps ($\alpha = \beta = 1$) on a Freudenthal mesh with 3 cells in each direction.
  • Figure 2: The $L^2$ de Rham complex in three dimensions (middle), and the finite element subcomplexes of the first (above) and second (below) kinds.
  • Figure 3: Hybrid subcomplexes of the $L^2$ de Rham complex.
  • Figure 4: Condition number of the diagonally-scaled stiffness and mass interior submatrices for $H(\mathop{\mathrm{grad}}\nolimits)$ on an equilateral reference simplex. The proposed basis in this work yields diagonal submatrices, while the sparsity-optimized hierarchical bases studied in beuchler2007sparse are less suitable for diagonal scaling.
  • Figure 5: (a) Freudenthal subdivision of a cube and (b) mesh of the Fichera corner.
  • ...and 3 more figures

Theorems & Definitions (18)

  • Lemma 3.1
  • Remark 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Remark 4.4
  • Lemma 4.5
  • Lemma 5.1
  • Lemma 5.2
  • Lemma 5.3
  • ...and 8 more